The quietest political election campaign in America is being fought in the U.S. Senate.

No bands, banners or bumper stickers for these candidates for one of the nation's most powerful offices. No glad-handing media events staged for television; indeed, news coverage is discouraged.It's the ultimate insider election.

The winner becomes the Democratic leader of the Senate, and if the party keeps its hold in the November elections, the winning candidate will become Senate majority leader.

The campaign, which has unofficially been under way for more than a year, began in earnest shortly before 2:30 p.m. last Tuesday when Robert C. Byrd of West Virginia announced he would not seek re-election as majority leader.

Within minutes, three senators let the press know they wanted the job. Others were said to have sniffed the winds, but were quietly discouraged by fellow Democrats.

Because the three who are running didn't call any formal news conferences, and because they won't even want to be seen as campaigning, brief profiles are in order:

Sen. Daniel K. Inouye of Hawaii. Inouye is 63, has served five terms in the Senate and is secretary of the Democratic caucus, the No. 3 position in the Senate Democratic leadership. Inouye was handed a plum last year when he was named chairman of the Select Committee on Secret Military Assistance to Iran and the Nicaraguan Opposition the man who ran the Senate's summer-long Iran-Contra hearings.

Inouye rates his chances as "very good" despite some quiet party criticism last year that he allowed Lt. Col. Oliver L. North to steal the show. Disclosure of his maneuvering to get money for religious schools in France included in a budget bill last December also hurt, but Inouye has a long record of distinguished service.

In 1973, he gained prominence as a member of the Watergate investigating committee. During World War II, he lost his right arm fighting in Italy. He holds the Distinguished Service Cross, the nation's second highest military decoration.

Sen. George J. Mitchell of Maine. At 54, Mitchell is the youngest of the three declared candidates, a federal court judge who resigned his lifetime appointment to the bench in 1979 to accept an appointment to the Senate. He won on his own in 1982, and is running again this year.

Mitchell is considered by some the favorite in the race for Democratic leader, although it's a contest with a tradition of surprises. (Byrd won the job in 1977 when Hubert Humphrey conceded defeat a week before the vote).

Mitchell gained favor in 1986 as chairman of the Democratic Senatorial Campaign Committee, a year the party re-claimed its majority in the Senate. Consequently, Mitchell was named to the prestigious Iran-Contra committee and was given the honorary position of deputy president pro tempore of the Senate.

Sen. J. Bennett Johnston of Louisiana. Johnston is 55, has been in the Senate since 1972 and is midway through his third term, serving as chairman of the Energy and Natural Resources Committee.