Big city mayors can stop hanging their heads in shame while holding out their hats for federal dole.

Cities, to be sure, have long been maligned and neglected. But the majority of Americans still expect them to serve as centers of commerce and culture, not only for their own residents but also for suburbanites and country folk.

This expectation gives cities the right to the country's unstinted support.

The view Americans hold of cities as opposed to suburbs, as one alternative, and small-towns and rural areas as another, was determined in what Louis Harris, the public opinion analyst, calls a "massive survey" of 7,000 adults across the nation.

The gist of the survey, recently completed for the Department of Housing and Urban Development, is that a clear majority think cities rate higher than suburbs on a long list of indispensable elements of American life. h

As Harris told a group of city planners and arts administrators in San Antonio last week, the survey makes clear that suburban residents are not as content with the suburban way of life, nor are they as convinced that city life is worse, as prevailing wisdom suggests.

For all their acres of free parking space, for all their unending commercial strips and Muzak-drenched shopping centers, their tinsely Tyson's Corners and flimsy White Flints, the suburbs have not replaced the variety and the refinements that an affluent people expects.

What with the search for fuel and values, we can count on the political and business leaders of suburbia to try harder. In the years to come there is bound to be a transformation, we seem to be heading toward "suburban renewal."

Unfortunately, only a very few architect-planners are thinking about it.One the few is Arthur Cotton Moore, who has designed some interesting schemes in Washington's Virginia and Maryland suburbs.

The Harris survey found that:

More people (39 percent) believe cities do better than suburbs or small-town rural areas in public services such as garbage collection, street maintenance, fire and police protection. The suburbs scored 39 percent;

49 percent believe cities have the best parks and playgrounds. Suburbs: 32 percent;

62 percent believe cities have the best shopping facilities. Suburbs: 28 percent;

67 percent believe cities have the best colleges and universities. Suburbs: 13 percent;

71 percent say cities offer the best employment opportunities. Suburbs: 13 percent;

73 percent believe cities have the best medical facilities. Suburbs: 16 percent;

77 percent believe cities have the best restaurants. Suburbs: 14 percent;

78 percent believe cities have the best selection of movie theaters. Suburbs: 12 percent;

81 percent believe cities have the best transportation. Suburbs: 8 percent;

90 percent believe cities have the best and most plays, museums, concerts, dance and other cultural opportunities. Suburbs: 4 percent.

Interesting.

But do suburban residents act on these beliefs? Do they actually make use of the cities' purported advantages?

Harris asked suburbanites to recount in detail how many of nine key activities they engage in and where they go to do so. The response, he said, "is among the most startling in the over 2,000 surveys we have conducted."

Dinning out: An even 50 percent go to the city and 50 percent stay in the suburbs.

Religious services: 44 percent go to the city.

Medical care: 52 percent go to the city.

Visiting friends: 47 percent go to the city.

Furniture and appliances: 46 percent buy them in the city.

Movies: 53 percent of all movies seen are seen in the city.

As Harris said, all this will not instantly dispel "the libel that cities are useless entities, serving out their time until finally they self-destruct." a

But added to the likely need to save fuel and live "closer in," these marked preferences obviously spell the further urbanization of surburbia. There will be pressure for more townhouses and apartments, fewer free-standing houses and less waste of land.

With the increase in childless, small households, it stands to reason that there will be more demand for restaurants, specialty foods and shops, laundromats, bars, cafes, tennis clubs and other recreation, arts-and-crafts workshop, adult education, neighborhood and community activities.

In all of this, there is likely to be greater stress on quality, individuality and urbanity not to say sophistication. The rising expectation in the suburbs, now that nearly everyone out there has progressed from hi-fi to videotape recorders, is for public as well as private amenity, and for basic accomplishments of urban civilization such as sidewalks, park benches, and attractive, interesting public spaces.

like generals, who always are cleverly preparing for the past war, suburban developers display great genius in copying last year's success. That is why, according to one estimate, 60 percent of them go bankrupt every year.

Except for building routine hospitals or court houses, few architects take any interest in suburbia. Among the exceptions is Moore, who first came to public attention a decade or so ago by skillfully blending the old and the new at Georgetown's Canal Square.

One of Moore's designs shows us a simple way to make suburbia more compact. On the vast parking lot of the somewhat seedy Shirlington, Va., shopping center, he would build a mini-new town. With the housing intergrated with the remodeled shopping mall, we would have a beehive of human activity in suburban Virginia's monotonous disco-scape.

In Arlington, Va., Moore and his firm propose to give cohesion to what should be the town center but is only the usual cacophony of billboards and unlovely commercial structures. At every intersection, Moore would incorporate the tangle of overhead wires, traffic lights and traffic signs into an elegant arch spanning the road. The cumulative effect of the road. The cumulative effect of the arches is an arcade. It is festive. It give a place identity.

The residents rejected the idea.

Along Rockville Pike, Moore proposes to hide the strip development mess behind a green berm flanking the motor road. This dike to contain a sea of suburban ugliness is to be planted with periwinkle and interrupted only to allow entrance and exit to the parking lots and stores behind it.

Filling in underused space and taming strip developments is only a part of the beginning suburban transformation. Another is the addition of further community services of all kinds. This, too has already begun. With school enrollments across the country having dropped anywhere from 10 to 40 percent, thousands of vacant school buildings are being converted into community centers.

The new centers offer recreation, adult education, vocational training and other services ever more urgently needed by the blue collar workers and newly arrived immigrants who are beginning to invade what until recently seemed exclusive, white middle-class turf.

The turf, come to think of it, is generally a lonely place. The front lawn isolates and acts as an invisible wall between the private house and the public road. The private automobiles everyone moves in have been called "isolation boxes." Only the recent expanded use of citizen-band radio relieves their loneliness. Suburbia has few, if any, places for spontaneous encounters among friends -- places, as Lewis Mumford put it, "where lovers can meet."

The 1980's, I venture to predict, will shake things up a bit -- and thus, perhaps, ease the identity crisis, the "me" obsession, and the real or fancied need for psychotherapy.