Limited Advancement Opportunities: It's pointed out several times that he is permanently stuck in his position as an associate at Crane, Poole & Schmidt due to his complete refusal to play by corporate rules, but he's simply too good and valuable as a trump card to fire him.
The Lost Lenore: His wife passed away just as he was deliberating over getting rid of her. Since then, he drowns his loneliness and grief in a parade of comely paralegals.
Nice Hat: Not necessarily always a hat, but Alan is particularly fond of wearing strange things on his head. Ornaments (that light up, no less), mistletoe, wigs, coonskin caps...Bunny ears...
The Only One: He's the go-to man when it comes to impossible cases, since his originality may be the thing to save the day. Also, a lot of such cases are so fucked that there's nothing he can do to screw it up even more.
Hidden Depths: He can be surprisingly insightful and emotional when caught in a quiet moment.
Hypocritical Humor: He claims to have never lost a case, and frequently proclaims "Still undefeated!" after a win. However, he has been shown to lose several cases, though he purposefully takes second chair on cases that he thinks might be losses so that he can claim that it was the first chair attorney, not himself, who earned the loss. Despite that, he still claims the victory on cases won with himself as second chair.
The Not-Love Interest: For Alan, to the degree that, when Denny decides he wants Alan to inherit his estate when he dies, they get married to seal the deal.
Obfuscating Stupidity: Depending on the writer if he was either completely out of his mind or just pretending out of both a sense of whimsy and a genuine belief that if he does, then when he really is past the point of no return, neither he nor anyone else will notice.
Screw the Rules, I'm Doing What's Right!: If you're a scumbag with AIDS who raped and murdered a little girl, Denny doesn't give a fuck about attorney/client privilege or any of that other crap. He'll proceed straight to shooting you in your kneecaps and suggest you get another attorney while you're lying on the floor writhing in pain.
Showy Invincible Hero: "Still undefeated!" Denny has an enormous ego and isn't afraid to show it, but his insane behavior in court only makes it more fun.
Straw Character: Denny Crane is a strawman political caricature of a conservative gun nut, although this did allow him to save the day a few times. Keeping guns in your office seems less crazy when you just used them to shoot the man about to kill your best friend.
Unusual Euphemism: Played for drama. He's in early stage Alzheimer, but refers to it as "Mad Cow" to take some of the edge of it. Alan is well aware and goes along as a show of kindness towards his friend.
The Bus Came Back: It never really left. He just moved to a position in the firm where he dealt less with the named partners when Carl transferred to Boston to act as Denny's handler. Paul seems to be happy about this when we see him next, and it's implied that he was burnt out from having to put out all the fires Denny started.
FaceHeel Turn: He arranges for the merger with the Chinese in the series finale.
HeelFace Revolving Door: When first introduced, he was clearly an antagonistic character, a foil for Alan and Denny both. He was not in any way sympathetic, and seemed to hate Denny so much that he was willing to hurt the firm to bring Denny down. Later he softened, became more sympathetic and took the role of Only Sane Man as well as a bit of Butt-Monkey.
Security Blanket: He has a wooden cigarette that he fidgets with when he needs to use up excess nervous energy (like when making a statement in court). He'll even take a puff, as if it were a real cigarette, when needs a major confidence boost. Denny once tempted him to turn on Alan by offering him a wooden cigar.
Extreme Doormat: His actual personality. He developed Clarice, Clavont, and the others as a way of being able to become someone else that didn't have his crippling shyness.
Grew a Spine: Starts off unable to be himself around others, using his invented personas, and refuses to even try to practice law. By the time he leaves the show, he has more than once defended a client in court and has developed the ability to stand up for himself.
Morality Pet: Claire genuinely likes him and tries to help him become more assertive and confident.