Beatrice Brown has a simple dream. "I would like to have a yard of my own where I could plant me a flower," she says. "And, oh, I'd love to have some rose bushes."

In all her 56 years she has never had a garden, never had a yard, never had anything, in fact, but a series of shabby tenements on this city's South and West Side neighborhoods. From her third floor apartment she can see grim high rises, old houses and vacant lots across the street. But she hides the grayness outside with her "lovely plants" inside - begonias, English ivy and the long-leafed mother-in-law's tongue, "or maybe it's called devil's tongue," she laughs.

She and her husband Ollie and five of their nine children have lived in the apartment about a year. They dream of finding a four-bedroom townhouse in the suburbs where, says Ollie Brown, "there's decent living, decent schools, not as much crime, and where, I believe, we can find some peace."

The problem is that they are poor and black and the suburbs are affluent and white.

Their dream is much like that of Mama, the leading character in "A Raisin in the Sun," who for years lovingly nourished a "raggedy-looking old thing" of a plant, which for most of her life was as close as she could get to having a little garden of her own. Beatrice Brown knows all about the heroine of the prize-winning play, which takes place in Southside Chicago. "Mama," she says proudly, "that's me."

Twelve years ago, when the Browns lived in the Henry Horner Homes public housing project on the West Side, they and five other black families sued the Chicago Housing Authority in an effort to get subsidized housing for minorities in white neighborhoods.

The suit, supported by the American Civil Liberties Union, charged CHA with racial discrimination in assigning tenants and locating public housing projects. The families also sued the federal Department of Housing and Urban Development, which funds CHA, alleging that it knew about and acquiesced in the discriminiation.

The court decisions fell first on CHA, then on HUD. In 1969 CHA was ordered to build more public housing in white areas than in black areas. In 1972 it was ordered to bypass the city council, which was rigidly refusing to permit public housing in white neighborhoods. In 1976 HUD was told it could provide relief to the plaintiffs not just within the city but in the whole metropolitan area.

The case, by this time callled Hills v. Gautreaus, was regarded as a virtual Magna Carta for minority families seeking low-cost housing. It was supposed to do for people in subsidized housing what Brown v. Board of Education did for black children in segregated schools. Edward L. Holmgren, director of the National Committee Against Discrimination in Housing, recalls that his organization "and the entire civil rights movement hailed the Gautreaux decision as the enunciation of a great and far-reaching principle of law."

The principle, basically, is that just because you are poor and black, that doesn't mean that you are barred from living anywhere but the slum ghettoes of the city. The reality, however, is almost exactly the opposite here - as it is in the rest of the country.

Since the 1969 ruling little subsidized housing has been provided for poor families anywhere in Chicago or its suburbs, and virtually no dispersal has occurred.

Of some 40,000 black families either in public housing or on waiting lists for it, only 410 families have moved to subsidized units in the suburbs under a special HUD-funded program that started two years ago. A few hundred others have moved to subsidized units in white areas of the city.

The reality of Ollie and Beatrice Brown is that they have been unable to find the subsidized four-bedroom townhouse in the suburbs that they want. To escape the crime, the rats and roaches and the constant powdery dust from the crumbling cement walls, which aggravated their children's asthma, they moved from the public housing project to their current apartment. It is not subsidized, and their monthly rent doubled to $185 plus utilities.

"The landlord keeps the bugs down, and our cats keeps the mice away," says Beatrice Brown. "And there's not as much crime. At least our daughter has not been beaten and robbed here like she was once at the project."

But Ollie Brown, a $7,200-a-year security guard at a local union hall, insists that "the neighborhood is not sage. There's nothing going for families. The community is going down." At 60, he hopes to retire soon on disability pay. "All my life I've lived in the city," he says. "I want to live out, where it's not so crowded. Chetto life means deteriorating buildings, deteriorating streets, deteriorating people."

OLLIE AND Beatrice Brown are pseudonyms. "Don't use our real names," the husbands pleads. "I feel if we told our names, some blacks would resent us." Indeed, their obvious ambivalence about the lawsuit, which has yet to bring them the housing they want, illustrates one of the great ironies in civil rights law: This case, and the failure surrounding it, has pitted poor against poor, black against black, white against white, and liberal against liberal.

While "Ollie Brown" says Gautreaux "was right for us and for others," Johnny Golden, who is also poor and black, is annoyed by it.

"They ought to get us some housing," says Golden, who works nights as a machine operator and whose wife works days as a city hall clerk. "I don't care about that court devision. I just ain't got no reason to go to the suburbs. My job is in the city. What do I want to go there for? The problem is nobody wants you if you got four kids." (Last week, after looking more than a year, Golden found a four-bedroom apartment in the city and plans to move in later this month.)

Rep. Cardiss Collins (D-Ill.), a West Side resident, is a black liberal who supports the open housing goal of Gautreaux but asks, "What good has it brought to black citizens?" She is locked in battle with another black liberal, HUD Secretary Patricia Roberts Harris.

Collins has introduced a bill that would allow HUD to ignore the Gautreaux ruling and approve new rent-subsidized housing in black areas of Chicago regardless of whether it goes in white areas. But Harris resuses to support the bill.

"I am not going to increase segregation in order to build new housing," Harris says flatly. "I do not believe it is legal or appropriate for the federal government to put its stamp of approval on racial segregation and say that just because we have to fight so hard for dispersal, we'll give up and provide funds only in places where the projects already are."

J.S. Fuerst, a white liberal who teaches urban affairs at Loyola University, charges that as a direct result of the Gautreaux decision, "the opportunities for good public housing are smaller than they were 10 years ago; prospects for new public housing in Chicago are now virtually nonexistent." Writing in a recent issue of the quarterly, The Public Interest, Fuerst places much of the blame on the principal lawyer for the Gautreaux plaintiffs, Alexander Polikoff. It was Polikoff who drew up the court's "Procrustean remedies for Chicago's complex housing problems," Fuerst charges.

Polikoff, also a white liberal, argues that the Gautreaux remedies were themselves compromises and allow for exceptions. Noting the city's and CHA's past record of discrimination, Polikoff accuses them of continuing to discriminate. "The city is stonewalling; CHA is stonewalling," he testified at a congressional hearing here last month. "They just don't want to take the political flak for supporting poor folk in white neighborhoods."

City officials would not comment on the charge, but CHA denies them, saying its failure to build is due to the fact that public housing costs more to construct than HUD is willing to spend and to the general unavailability of sites in white areas. Yet at the hearing, chaired by Collins, they admitted finding 500 potential sites but building on only 25 of them, a record Collins called "lousy."

HUD's deputy general counsel, Edward W. Norton, agrees with Collins that assisted housing production "is well short of what is needed," but he stresses that HUD does not believe Gautreaux is to blame.

CHICAGO'S failure is tied to the failure of housing integration generally. Despite passage of the national Fair Housing Act 10 years ago and despite the integration of many neighborhoods since then, most of the country is still tightly segregated. Chicago itself, which pioneered racially restrictive covenants in the 1920s, is still known in fair housing circles as the most segregated city in the country.

The failure to disperse low-income family housing here and elsewhere is even more acute because middle-class people - black and white - simply do not want it nearby. They see the issue in terms of them verus us. At times they feel racial prejudice and, almost always, class prejudice. Their overriding emotion is fear. Fear that an influx of them - poor welfares types - will drive down our property values. Fear that they will strew beer cans all over our neighborhood. Fear that they will commit crimes on us :

Never mind that most of the poor are every bit as worried as most of the affluent are about property, litter and crime. It's the troublemakers that count. And the fact is that we don't want them .

Here in Chicago, there is no doubt that Gautreaux has complicated the matter. In 1969 the federal district judge in the case, the late Richard Austin, allowed the Chicago Housing Authority to build 1,458 apartments then being planned in black areas of the city. But he ruled that the next 700 units had to build in white areas and that from then on public housing construction should proceed on a three-for-one basis - three units in white areas for every one built in a black area.

After Congress created the rent subsidy program under Section 8 of the 1974 Housing and Community Development Act as the primary federal tool to provide housing to low and moderate-income people, the court order was amended to say that units under the new program should be built on a 60-40 basis - 60 units in white areas for every 40 in black areas. Austin defined black areas as those with a black population of 30 percent or more.

There could be exception to his order, Austin said, but the CHA sought very few and the city did little to encourage private developers to provide Section 8 housing. In addition, Richard Nixon's 1973 moratorium on subsidized housing held down construction all over the country for at least two years.

As a result, in this city - where an estimated 200,000 families and individuals need housing assistance - only 184 public housing units and only 467 new Section 8 units have been built and only 5,454 units in all - new and existing - have been provided for lower-income families in nine years. In the same period 8,192 units have been provided for elderly.

In the six-county suburban area, production has been just as bleak - 5,955 units for families and 8,107 for the elderly over nine years.

FUERST HAS called Gautreaux "a toothless gesture and nothing more" in its ruling that HUD could use the suburbs to promote dispersal of low-income housing. The Supreme Court said that housing problems are metropolitan problems - but it added that no suburb could be forced to take any HUD housing it did not want.

So Polikoff and HUD were left to fashion a remedy. They came up with a demonstration project in which a fair housing group called the Leadership Council for Metropolitan Open Communities would find landlords in the suburbs willing to rent to blacks in, or eligible for, Chicago public housing.

The tenants, who are screened by the council and moved only if the new landlords agree, pay a quarter of their income in rent and HUD, using Section 8 funds, sends the landlord the difference between what the tenants pay and what the fair market rent is.

Kale Williams, the council's staff director, says 460 families have moved so far, 50 of them within the city. The families are in 110 developments in 65 communities. Eleven families have quit the program voluntarily and 13 others have been evicted or not accepted for renewal. The failure rate is very small, says Williams, "and we find that these families make good tenants. The landlords are satisified."

However, he admits that "obviously we're not satisfied with the rate of placement. It's 20 to 25 a month - a minuscule number - but it becomes more impressive if you compare it with what most housing authorities are doing across the country." The experiment is funded until Jan. 15; and by the end of the year HUD must decided whether to continue it, expand it, try it with another fair housing group or scrap it.

Fuerst contends the HUD experiment is the wrong way to achieve suburban integration. "You don't use Gautreaux families as the battering ram or leading phalanx," he argues. "Gautreaux will succeed only when middle-class whites are willing to accept middle-class blacks. Only after that can you introduce the lower economic spectrum. They're really a ver small number, but people in the suburbs think they're all that there is."

Fuerst may be right about prevaling middle-class attitudes, but his prescription runs counter to national policy. The 1974 Housing and Community Development Act clearly states that one of its objectives is to reduce the concentration of the poor in central cities.

His solution also ignores Supreme Court rulings going back at least to 1917 (in a housing segregation case) and repeated in 1955 and 1958 school desegregation cases. Those ruling stress that the "vitality" of constitutional rights "cannot be allowed to yield simply because of disagreement with them."

BUT ENUNCIATION of federal policy and court decisions alone cannot dent the armadillo resistance to dispersal of low-income housing.

Polikoff, in a book called "Housing the Poor: The Case for Herosim," argues for a new law giving the federal government power to allocate such housing, overriding local zoning laws if necessary, and to build the housing itself if no one else will. He concedes the idea is not "politically feasible" now but says it might be, given "presidential leadership."

Jimmy Carter hardly seems the leader to take on this one, and Congress is not exactly on a civil rights rampage. What might work is more toughness on the part of HUD. The 1974 law specifically links a community's performance in providing housing for the poor to its eligibility for community development grants.

HUD has begun to tell communities, including politically potent giants like Chicago, that they must meet the housing needs of lower-income people or risk losing community development funds. What is needed now is for HUD to impose more exact conditions and stricter timetables.

Critics argue that if HUD pushes too hard, localities will simply tell the government to keep its money. Maybe. But with taxpayers in revolt, local officials are increasingly reluctant to let any revenue source - especially the generous federal government - slip away. Chicago, for example, is getting $115 million in federal development money this year.

HUD could also urge the Justice Department to bring more fair housing suits. But some HUD officials fear that Congress may rein in the agency if it appears overzealous. "We get no pressure from anyone in Congress to do more and a gread deal of pressure to do less," says one.

Secretary Harris, however, seems willing to push. "HUD could do more," she admits. "We're working quietly with jurisdictions to take low- and moderate-income housing, but the pace is too slow. We could assist fair housing groups to work more with local communities.

"We're going to crack down on the 5 percent of subsidized housing where there has been bad management. There's nothing that says these projects have to be crime-ridden or rundown. We've started a program of more than $200 million to upgrade management and living conditions in the worst of the projects. And we're encouraging authorities and private developers to build smaller, low-rise projects that are scattered and blend in with the community."

HUD is pushing production. It has began to overcome the effects of the housing moratorium, the past low funding of subsidized housing construction and the financing problems that developers initially encountered in building Section 8 units. While only 12,000 public housing units were started in the fiscal year that just ended, some 35,000 starts are planned for the coming year. More than 140,000 Section 8 units were started last year, and about 180,000 are planned in fiscal 1979.

Whether stepped-up construction and heavier government pressure on suburbs to accept the units will affect people like "Ollie and Beatrice Brown" is anyone's guess.

They remain optimistic. "We're hoping to get out of here next spring," he says. "We'd like to go the northern suburbs. Evanston seems to be opening up to blacks. Maybe they'd accept us." His wife recalls the story of Mama in "A Raisin in the Sun:" "She did make it, you know. In the end, she got out of the ghetto."