And wouldn't you know, he could act too. Three of the best:
From Here to Eternity (1953). The role that saved Sinatra's stalled career (and spawned the horse's head incident in The Godfather) is doomed Pvt. Maggio. The movie is a bittersweet classic, from Burt and Deborah Kerr's seaweed scene to a lot of Montgomery Clift's best stuff. Oscars galore. See it again.
The Manchurian Candidate (1962). John Frankenheimer directs Sinatra, Laurence Harvey and Angela Lansbury in a political chiller. Savvy and memorable dissection of anti-Communist paranoia in Washington; the garden-party scene is art.
Ocean's Eleven (1960). The Rat Pack. Rat Pack movies are tough. Frank, Dean, Sammy, Lawford and Bishop: you try so hard to love them as much as they loved themselves. Ocean's Eleven is the prototype, and probably the best of them. Good plot (army buddies knock over five Vegas casinos), and Cesar Romero is great icing as kingpin Duke Santos. But it's stiffly paced and self-indulgent. Go ahead and see it anyway.
There are more, so many more, but these three crown the canon. Grab one, tilt your fedora, and pay tribute — while you can still feel good about it.