The announcement yesterday that the largest crowd in inaugural history -- 140,000 strong -- will be invited to watch President Reagan sworn in Jan. 21 came none too soon in status-conscious Washington, where the scramble for the almighty inaugural ticket is already under way.

There are tickets to balls, tickets to galas, tickets to parades and tickets to the ceremony. There are tickets to sit and tickets to stand -- each with its own price and each with its own measure of prestige.

The catch is that for everything but the parade, you need an invitation to buy. Money alone will not do. Thus, the "hottest ticket in town" is to the Frank Sinatra-produced presidential gala Jan 19. There are only 6,000 to go around, at up to $200 a head, and those without invitations have already started dialing anybody they know with presidential pull.

"Things haven't gotten ugly yet," said one White House aide, "but I'm sure they will."

Sen. Charles McC. Mathias Jr. (R-Md.), chairman of the congressional inaugural committee, announced details for the ceremony yesterday. But a Mathias aide said he already has gotten calls from a baroness in New York, a school teacher in Ohio, and a lobbyist who wants tickets for his clients in the Mariana Islands. "The Fortune 500 are well represented in my callback slips," said John Chambers. "I call everybody back . . . and tell them I can't help them. They've got to see their congressman or senator."

More than 72,000 of the total tickets for the ceremony -- all of which are free -- are parceled out in packets of 230 to senators and 120 to members of the House. The allocation system knows no political lines. Republicans and Democrats get the same size packets.

One Democratic senator has promised his Republican colleague all but a handful of his allocation. But he wants the favor kept quiet, his aides say. It wouldn't fit the image of the battling duo his constituents expect.

As in all things political, the allocation system -- in which some members have tickets to spare and others must go begging -- has given rise to some resourceful dealing on the Hill.

Take, for example, Rep. Cardiss Collins, a Chicago Democrat whose constituents have little interest in the Reagan inauguration, but just love the pretty congressional calendars she sends to them each year. The problem is she only gets 2,400 calendars.

So Collins recently sent out what's known as a "Dear Colleague letter" offering a simple trade. "We've had quite a few takers," said Bud Myers, one of her aides. Collins' going price is 30 calendars per ticket.

The constituents who ultimately receive the tickets will see Ronald Reagan sworn in as president of the United States in a ceremony with some of the heaviest security in history. For the first time, every guest who enters the huge fenced-in inaugural area on the Capitol's West Front will be funneled through a metal detector, a procedure that has become the norm for presidential events since Reagan was wounded in a 1981 assassination attempt, authorities said.

Mathias said that thousands more will be able to gather on the mall to hear the ceremony from speakers that will be set up.

Reagan will actually be sworn in Sunday Jan. 20, the day mandated by the Constitution, in what aides have called "a very private, very personal" ceremony in the White House. But following in the footsteps of previous presidents whose inauguration day has fallen on Sunday, he will repeat the oath publicly on Monday, then lead the traditional inaugural parade down Pennsylvania Avenue.

The weekend will be filled with free fireworks spectaculars and a "young Americans pageant" for the public and the series of invitation-only events that have set phones jangling over at inaugural headquarters.

Staffers say that about 50,000 invitations to balls and 12,000 to the two galas, including one Jan. 18 for the vice president, "were out the door" as of yesterday.

"After Christmas the ugly realization will dawn on some people that they haven't been invited," said one White House aide. "Then there'll really be heat."

But it is already getting warm in some offices at 1600 Pennsylvania Ave.

Sheila Tate, Nancy Reagan's press secretary, said she got a call from a "frantic columnist" who got six tickets to the presidential gala in 1981 and counted on them again this year. He's already promised them to houseguests but so far he hasn't been invited, she said. "I told him it's being handled so fairly that there's nothing I can do."

On the Hill, however, some are willing to be more resourceful. One aide confided that some members of Congress, when filling out their "A list" for ball tickets, put on the names of several staffers who don't plan to attend. Then they can parcel out those tickets at will.

And in the office of Rep. Frank Wolf (R-Va.), whose suburban Washington district leaves him deluged with ticket requests, one aide promised, "Before it's over we'll fill every request, or at least we'll die trying."