How closely connected to organized crime families was actually Frank Sinatra?

James Kaplan's lengthy two-part biography of Sinatra (2010's Frank: The Voice and 2015's Sinatra: The Chairman) makes the case that Frank Sinatra wasn't a puppet of the mob, but that he 'unfortunately' very much admired them and thus had extensive contact with them. Kaplan argues in Sinatra: The Chairman that:

Sinatra's Mob associations had far more to do with mutual admiration than affiliation. The gangsters liked his singing, his flash, and (at times) his unrepentant unruliness; he liked their power, their toughness, their swaggering style. Growing up in an era when power was largely in the hands of white Protestant men, a time when Italian-Americans were just a half step up the social ladder from African-Americans--and, like black people, were seen as simple, happy, and musical--Frank viewed the Mafia as a kind of unelected elect, an alternate aristocracy. He idolized them all his life, much as a small boy might idolize cowboys or soldiers.

I mean, take the Frank Sinatra album Sinatra At The Sands, a live album recorded at The Sands Hotel and Casino in Las Vegas, released in 1966. As this Vegas mob ownership answer by /u/grandissimo points out, Vegas casinos were, during the time period, widely suspected of being mob fronts. In 1954, slightly before Sinatra become an adult popular music megastar with the likes of Swing Easy and Songs For Young Lovers, Joseph Stacher, a New Jersey crime lieutenant, had supplied Sinatra with $54,000 in order to buy two points of ownership in the Sands. The Nevada Tax Commission was extremely dubious that this was actually Sinatra's own money, and they challenged Sinatra's purchase of those two points for two years before mysteriously caving. By the beginning of 1958, Sinatra owned nine points in the Casino, which Kaplan credits, 'some say [to] a gift from the mobster Vincent 'Jimmy Blue Eyes' Alo'.

Kaplan seems to imply that these weren't so much Sinatra being an active participant in washing dirty money as the mobsters liking Sinatra and so actually giving him gifts. Sinatra received a healthy monthly check from The Sands, but likely that was only a tiny fraction of the amount of money he would have received had he really owned nine points in The Sands.

He quotes Sinatra's daughter Tina, who told Seymour Hersh that:

By the thirties and forties when Dad was in the business, they were controlling the nightclubs, they were controlling the entertainment world. They were a motivated bunch. The power of an entertainer and the power of a mobster--it's all very much a part of America. They were all from the same neighborhoods. My Dad grew up with gangsters next door. He was living with them. They were his personal friends, and he's not going to cast away a friend. The great vein through Frank Sinatra is loyalty. There is an absolute commitment to friends and family. It's very Italian and probably gave him a little more in common with the mob types.

Kaplan argues that the Mob basically tolerated Sinatra, occasionally found him to be useful (e.g., with the Sands), and that because he was a fellow traveller, if not a member or a bought man, they threw him a bone.

Edit: Perhaps the most famous story of Sinatra’s involvement in the Mob is fictional, however, according to Kaplan; The Godfather of course has a lightly fictionalised version of Frank Sinatra, Johnny Fontane, appealing to Don Corleone to get him a part in an upcoming film, involving the infamous horse head in the bed. This is widely taken to refer to Sinatra surprisingly getting a role in To Here To Eternity over Eli Wallach, who was the movie producer’s first choice for the role. However Kaplan argues that the reason why Wallach didn’t get the role was that Wallach wanted too much money to play the part, and that Sinatra - who was obsessed with the book and desperately wanted the role - would have done it for peanuts. No horse heads involved!