HOLLYWOOD -- Frank Sinatra is back on cloud eight, bouncing around as jauntily as ever. Outwardly, at least, he shows no signs of the buffeting he has weathered since 1951 when his career appeared to be slipping away. The Hoboken minstrel is made of stern stuff and Hollywood, which, like the rest of the world, admires a fighting spirit, is looking up to him again. At the age of 37 he is off to a fresh screen career - one which some producers believe can carry Sinatra as far as he wants to go.

Nobody helped him to climb back on the Hollywood bandwagon from which he toppled rather unceremoniously with ''Meet Danny Wilson.'' That Universal film, aside from its dramatic shortcomings, had the misfortune to be released at the time when Sinatra was riding a wave of bad publicity and the fans appeared to be deserting him in droves. The story of his remarkable comeback as Private Maggio in ''From Here to Eternity'' has been pretty well told since Sinatra sprinted on stage at the Pantages Theatre before the watchful eyes of television reviewers to accept the Oscar fro best supporting actor performance in 1953.

Since ''Eternity'' Sinatra finds himself having to remind producers that he's a song-and-dance man. His plan to alternate between roles in dramatic and musical films was unexpectedly upset when Marilyn Monroe walked out of ''Pink Tights.'' As a result, Sinatra currently is doing a second dramatic role, that of a hired gunman in Robert Bassler's independent production cryptically titled ''Suddenly.''

There is no hint of the melody man in this characterization of a former soldier who hires out to a group which offers him $500,000 to assassinate the president of the United States. But Sinatra's pipes are in fine shape and he can warble a popular tune with the same ease which convulsed the bobbysoxers in the forties. His records are back in the big-seller lists.

''Sure I was so mixed up then,'' he said, referring to the personal and professional problems which closed in on him four years ago, ''that my singing was affected. I knew it. I offered a couple of times to pay for the recording session myself if they'd agree to hold back the records. Nobody has to tell me when I'm not singing. I don't have to wait to find our here,'' he said, pointing to his ears. ''I can tell if a note is going to be sour when its down here,'' he illustrated, jabbing himself in the diaphragm.

Sinatra always has had the reputation of being a conscientious worker. He tenses, like a fighter going into the first round, when he has a job to do. But the tension is caused by concentration, not by uncertainty. This became evident as one watched him doing scenes for ''Suddenly.'' When the action was over, his whole body seemed to melt into relaxation.

His concern for his own ''safety'' as well as that of the picture is manifest in ways which could easily lead the casual observer to accuse him of being difficult. Mr. Bassler confesses to having experienced a feeling which might be described as ''dread'' when Sinatra came to him to discuss changes which he felt would benefit the film. ''I steeled myself, for there is nothing more disturbing than the cerebrations of an actor,'' the producer commented. ''But Frank wasn't making demands to exploit himself at the expense of the picture. The suggestions he offered made sense.''

The Sinatra temperament bursts into fiery flame, however, when gossip columnists pry into his personal life. The break-up of his marriage to Nancy Sinatra, mother of his three children, and his present union with Ava Gardner are matters which he feels should not be discussed in the public prints. Those are the dark clouds hovering over what otherwise appears to be a bright future. ''I have to be careful not to carry myself too far, too soon,'' the ''Voice'' said in thoughtful contemplation of the many offers he now has. There is in the back of his mind, he admitted, a determination to gradually drift behind the camera and become a director.

Presently his film career is going in several directions. He has agreed to an offer from Stanley Kramer to team with Robert Mitchum in ''Not As a Stranger,'' another non-musical performance. He is considering a proposal by Warner Brothers to go into ''Someone to Watch Over Me,'' a new adaptation of ''Four Daughters,'' in which he would assume the role that brought the late John Garfield to the attention of moviegoers in 1938. He has a commitment with Metro to costar with Miss Gardner in ''St. Louis Woman.'' Meanwhile, that studio is trying to work up a musical which would bring Sinatra and Gene Kelly together again. The boys, it will be remembered, went well together in ''Anchors Aweigh,'' ''On the Town,'' and ''Take Me Out to the Ball Game.'' Finally, he may be doing a picture for Twentieth Century-Fox which extended through next month the contract it made with him for ''Pink Tights.'' The script of the musical is undergoing a rewrite, and Sinatra may wind up doing it after all.

So much for his Hollywood affairs at the moment. Although his memories of television are bittersweet, he intends to go back into it. The time, however, he won't get involved in a weekly one-hour show. ''A half hour sounds inviting,'' he said. ''Have you got any ideas for a suspense-type program?'' The new Sinatra definitely is a man who wants variety, but, he adds, ''I'm still a song-and-dance man. That's what I know best. I like to sing. That's why I like to play night clubs. It's fun.'' Profitable, too, considering that ''Frankie Boy'' had moved into the million-dollar-a-year bracket in 1944, three years after he became a headliner.