"I Want" Song: "Where Is Love" is him singing about wanting love.
Pinball Protagonist: He's an orphan, gets passed from orphanage, to a funeral home, then gets kicked out and gets picked up by the thieves guild, then is taken in by a rich old man. It's a musical, and the characters mostly sing around him as well.
"Consider yourself at home, consider yourself one of the family."
Get Out!: When Bills shows up with blood on his coat, a horrified Fagin realises what he's done and yells at him to leave.
HeelFace Turn: Subverted in the movie. He plans to do this, but instead chooses to leave with Dodger and continue a life of pickpocketing. It's rather heartwarming, in a strange way. In the theatre version, however, he plays it straight, deciding that with the breaking up and arrest of his gang, along with the loss of his precious treasures, there has never been a better time to change his ways.
Karma Houdini: At the end, even though he does lose his treasure he and Dodger are still free.
Lovable Rogue: De-emphasizing his Greedy Jew characterization in the original. Particularly notable is Ron Moody's cheerfully hammy performance in the 1968 film version. (He even stands up for Oliver on two occasions.)
Noble Demon: He's a thief, who trains others to be thieves, and works with Bill Sykes, but deep down he doesn't like his life and wishes the things he does weren't necessary, and tries his best not to be too bad. This trope is exemplified in the song "Reviewing the Situation":
Though I'd be the first one to say that I wasn't a saint, I'm finding it hard to be really as black as they paint. I'm reviewing the situation; can a fellow be a villain all his life?... I'm a bad 'un and a bad 'un I shall stay. You'll be seeing no transformation, but it's wrong to be a rogue in every way. I don't want nobody hurt for me or made to do the dirt for me, this rotten life is not for me, it's getting far too hot for me, there is no in-between for me, but who will change the scene for me? I think I'd better think it out again!
Tragic Villain: A variation. "Reviewing the Situation" shows Fagin as wanting out of the criminal life, but feels trapped by it. By the end of the film, he more or less accepts his lot in life and seeks to rebuild. In the play, he instead resolves to turn over a new leaf, and limps away.
Villain Song: "Pick a Pocket Or Two". "Reviewing The Situation" is a rather sad kind because it's him singing about how he wants out of the criminal life but feels trapped in it.
Disproportionate Retribution: Speaking his name aloud is, apparently, grounds for being killed, even if it's a whisper. He claims he actually followed through and did kill someone for boasting that he could his name in vain.
Never My Fault: In the film, after he murders Nancy, he's heard saying to himself, "Nancy, I loved you, didn't I? Look what you've do to me?!" How do you figure that one, Bill?
Sanity Slippage: After he kills Nancy, he becomes rather unhinged, to the point even his own dog ditches him.
Ungrateful Bastard: Nancy loved him enough to keep him out of Jail and he still goes and kills her.
"Oom-pah-pah, oom-pah-pah, there's a little ditty, they're singing in the city, especially when they've been on the gin or the beer, if you've got the patience, your own imagination, we'll tell just exactly what you to hear..."
All Girls Want Bad Boys: She sings about her love of Bill Sikes, the best thief, who's a cold, bullying monster. It turns out bad for her in the end - he winds up killing her.
Broken Bird: Her plight is summed up in her song, "As Long as He Needs Me" which explains why she stays with Sikes despite his domestic abuse.
Indy Ploy: How she manages to get Oliver to the bridge - she starts up a huge song and dance number to cause a distraction, then grabs him and runs out. Too bad Bull's Eye blew her cover...
Love Martyr: She has this BAD for Bill Sykes. She recognises this in "As Long As He Needs Me", but even though he's a murderous thug and robber who beats her and plans to kill Oliver, she still can't bring herself to hand him over to the law.
As long as life is long I'll love him, right or wrong And somehow I'll be strong As long as he needs me.