"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."
Affably Evil: He gets a charismatic song to introduce himself, and his pickpocketing is shamelessly cool. Not to mention he's clearly learned nothing by the end, and even uses the distraction of Bill's public hanging to pick a couple more pockets.
"In this life, one thing counts, in the bank, large amounts,"
Actor Allusion: At one point in the 2012 production, Fagin (played by Neil Morrissey) says "Can we fix it?" while playing around with his treasure.
Adaptational Comic Relief: While Fagin remains an unrepentant thief, he is a much more sympathetic and comic character than he is in the novel. His plot with Monks is deleted and his role in Nancy's death is similarly excised, and he is portrayed as being cowardly and deeply afraid of Bill Sikes.
Adaptational Heroism: While still a pickpocket, he's more of a sympathetic character in the musical compared to his portrayal in the original book. In the book, he makes Bill think that Nancy informed on him (when it was precisely the opposite). In the 1968 film, when he sees Bill chasing after Nancy, he calls out "please, Bill, no violence" and when he discovers Nancy's murder, he's horrified.
Adaptational Nice Guy: He's still not exactly a heroic character, being self-serving and at least somewhat manipulative. However, he's much nicer than his villainous book counterpart, as he does show genuine kindness and care to the children he trains as thieves and makes a dramatic contrast to the abusive and cruel adults Oliver has encountered previously.
Bad Samaritan: Downplayed from the book. While still somewhat self-serving, he is portrayed as something of a whimsical Loveable Rogue who does to some extent care for his gang (or at least loathes Sykes' treatment of them).
Even Evil Has Standards: He's extremely prejudiced against the way Sykes treats Nancy and the way Sykes uses violence to solve every situation.
Cool Old Guy: Often portrayed as an older man and something of a fun father figure to the boys.
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.
HeelFace Turn: Sikes forces her to lead Oliver into a trap so he can be brought back to Fagin. Later she risks her own life to save Oliver from Sikes.
Hooker with a Heart of Gold: Possibly one of the most famous examples in literature. It should be noted that 'prostitute' in Dickens's day just meant a woman who lived unmarried with a man outside of respectable society. So it's possible that's what the author was referring to in a preface (as Nancy only says she's a thief in the text).
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.
Unwitting Instigator of Doom: Oliver recognising her on the street and calling her by her name is what allows her and Nancy to pull of the scam that he's one of them.
The Voiceless: For someone who appears in several scenes, and gets prominently featured in most of them, she doesn't have a single line of spoken dialogue (but does sing solos in "It's A Fine Life" and "I'd Do Anything").
What Happened to the Mouse?: In the book it's said that she was so traumatised by hearing how Nancy died, she ended up in a mental hospital. She just kind of disappears towards the end, though some productions have her be among those who identify Bill's murder of Nancy.