All of the Other Reindeer: He suffers the feelings of this when he feels that nobody likes him because he's an Ogre, not a "handsome prince." This exchange sums it up quite well:
Donkey: What's your problem with the whole world?
Shrek: I'm not the one with the problem. It's the world that seems to have a problem with me! People take one look at me and go "AAH! Help, run! A big, stupid, ugly ogre!" *sigh* They judge me before they even know me. That's why I'm better off alone.
Terrifying Rescuer: Until Fiona realizes he's actually a much better prospect than the prince.
Villain Protagonist: In the first film, all he wanted to do was kick all the refugees out of the swamp by dealing with Farquaad's demands. Eventually, he grows out of this and ends up becoming The Hero.
Beware the Nice Ones: He is tolerant to Shrek's insults throughout most of the movie, but he has it towards the end and calls him out on it; but it's mostly because Shrek's missing his chance to tell Fiona how he feels.
The Pollyanna: If his stories are to be believed, he hasn't been very well treated in the past. Despite this, he shows pretty much entusiasm over everything and is never put down by Shrek's sarcastic insults. This could be that he's Sarcasm-Blind, but the calling out he does to Shrek at the end of the first film could imply that he's fully aware of Shrek's bad demeanor; he just chooses to ignore it.
Dating What Daddy Hates: In the second film, Shrek correctly predicts the hostility her father will show him for being an ogre. He gets better, though.
Et Tu, Rapunzel?: In Shrek the Third, the betrayal of her and the other Princesses by Rapunzel, who was in love with Prince Charming.
Everything's Better with Princesses: Both subverted and played straight. Notably, Fiona's perfectly willing to live in squalor with the man she loves, but she's got more of a sense for duty and family - and she's a strong feminist woman to boot.
Expository Hairstyle Change: While the hairstyle change itself isn't seen in-universe, she wears her hair down in Shrek Forever After's alternate universe as part of the ogre warrior clan.
Ambiguously Bi: In the second movie, when he hears the Ugly Stepsister describe how handsome Prince Charming is, he replies with "Mmm. Sounds dreamy."
Anti-Hero: When he first appeared he had no qualms about murdering Shrek for money. However, he shows a clear sense of ethics by joining Shrek in gratitude for saving him and quickly becomes a good guy for the rest of the series.
An Arm and a Leg: In the first movie, Farquaad tears off both of his legs, crumbling one to powder. The remaining leg is grafted back on, and by the time of the second film, he's gotten a new replacement leg.
Be Careful What You Wish For: Was probably not meant as An Aesop, and merely a horrifying ghost story to scare Shrek, but in Ginger Bride, he was kicked out by his girlfriend and asked the baker to bake him a ginger bride. He wanted too much sugar so she would be sweet and never leave him, despite warnings from the baker that it could have consistent side effects. When Ginger Bride later wakes up, she turns out to be pleasant at first, but slowly seems more attached to him and eventually becomes Ax-Crazy by desperation of wanting him, and more catastrophes are followed. This would never have happened if Gingy hadn't wished for too much sugar in her.
Running Gag: He always seems to come close to becoming a real boy, only for those plans to fall short, like in Shrek 2, when he does become a boy, only to immediately be turned back by accident, and in Forever After, when he's about to sign a contract to be turned into a real boy, only for the interaction to be interrupted.
Literal Ass Kissing: Unintentionally invoked. She was preparing to kiss Donkey (with the latter being somewhat unwilling). Shrek knocks Donkey off her, only to end up being kissed himself, right on the place that you'd expect. As soon as she found out what happened, she wasn't happy.
Our Dragons Are Different: Initially appears to be your average guardian fire-breather. Turns out she's sapient and much more developed than classic fairytale dragons, although she doesn't speak, and she can interact with society to an extent. Also, has a thing for talking donkeys.
What Happened to the Mouse?: The ending of Shrek 2 showed six Dronkeys including a red-colored one - named "Eclair" according to merchandise. Though, for some reason she never shows up in the third movie.
Villainous Breakdown: Has one when Shrek storms his wedding and Fiona reveals her nature as an ogress. He orders his knights to seize them both and starts shouting for order and ranting about how he's king.
Black Mail/Cold-Blooded Torture: Subverted only to be played straight later on: When she first appears to Harold, she implies that she is going to torture him when shouting at him for Fiona marrying Shrek. However, it immediately shows that the thing she was gonna do that she didn't want to if he didn't tell her was breaking her diet (apparently, she overindulges on junk food when irritated). Played straight, however, when she threatens Harold with what is heavily implied to be turning him back into a frog if he didn't spike Fiona's tea with a love potion.
Card-Carrying Villain: She's not ashamed to admit to the king that she uses tricks up her sleeve to manipulate people's feelings and make them fall in love, as she plans to do with Prince Charming and Fiona.
Fantastic Racism: An interesting example, her prejudice seems to be against people's level of physical attractiveness—sort of an extreme form of Beauty Equals Goodness. In her eyes, only beautiful people (or people who have been made beautiful) get happy endings. If they're ugly, then not only do they not get happy endings—but in her eyes, they don't even deserve happy endings.
Ambiguously Gay: Being voiced by the openly gay Rupert Everett doesn't help. There is a huge Parental Bonus in the scene where he talks to his mother about his reluctance to marry Fiona — apparently, he is not interested in women at all...
He has, however, shown revulsion towards the wolf when he sees him in what is supposed to be Fiona's bed, and it is also implied that he has some reciprocation of romantic feelings for Rapunzel.
Psychopathic Manchild: Hinted at in Shrek 2 (For one thing, when Fairy Godmother is forced to break her diet, he wanted and got a kids meal, shown to be a toy battle axe), and explicitly shown in Shrek the Third.
Shout-Out: His overall appearance, and to a lesser extent, his voice, is somewhat similar to Liquid Snake in renaissance clothing.
Overprotective Dad: He loves Fiona, but he doesn't seem to understand her, thinking she'd be better off with Prince Charming than an ogre like Shrek. He comes around at the last minute and accepts that Shrek is right for her.
Pinocchio Syndrome: His apparent backstory. Fairy Godmother turned him from a frog to a human so he could fall in love with his future wife.
Politically Incorrect Villain: His disapproval of Shrek for being an ogre is a metaphor for how some parents feel about their daughters dating outside their race.