We'll never really know. The scandal-mongers would like to believe that he was a Mob tool. It's undeniable that Sinatra liked hanging out with Mob guys, and that the places he played in Nevada had Mob money invested in them. So did that Westchester theater where he was photographed with Gambino and the rest of them. It's also widely believed that Willie Moretti helped Sinatra out of his contract with Tommy Dorsey, and that Sinatra sang at Moretti's daughter's wedding.
But the only place where Sinatra had his own money invested was the Cal-Nevada Lodge--and he was forced out by the Nevada Gaming Commission because he'd hosted Sam Giancana after Momo's name was added to the Commission's "Black Book." And, for that matter, every entertainer who appeared at any Nevada hotel (and plenty of other venues--the entertainment industry is notoriously Mob-dominated) at one time or another could have said to have been "connected" to the Mob via guilt by association. I'm guessing that the Mob didn't "order" Sinatra around. He always fancied himself as a "tough guy," and I'm guessing that he liked being around Mob guys, when and where it suited him. In his autobiography, Fat Vinny Teresa says of Sinatra: "He picks his spots to be a tough guy with people." Teresa tells how Sinatra, during his hard times before "From Here to Eternity" re-ignited his career, used to "beg" for work in Mob-controlled nightclubs. In one case, he asked the Mob operator of the Copa in Boston for a loan that the guy could deduct from his pay the next time he played the club. The movie came out and Sinatra was big again. He paid the Mob guy back but never played the club again because he didn't need to.