So, we begin with Maximus from Tangled, arguably one of the best of the lot.
We meet Maximus for the first time as a determined mount for the Captain of the Guard chasing Flynn Rider. He ditches the captain pretty quickly and enters into his own pursuit. His facial expressions are amazing and I'm most impressed at the way he still moves (mostly) like a real horse, despite being way more anthropomorphic than the average cartoon horse.
He quickly develops into a strong personality, taking his role as a guard horse very seriously and pursuing Flynn with determination, grit, and real anger. His actions are played as hilarious but the movie clearly respects him and just lets him be awesome. Even though you're supposed to sympathize with Flynn and Rapunzel, if you're a horse person, you're secretly (or not so secretly!) rooting for Maximus all along.
Eventually, he teams up with the heroes. He hates Flynn until pretty much the last few minutes of the movie, but he warms up to Rapunzel all unicorn-like. (Or maybe you're supposed to think that since he's a palace horse he somehow recognizes that she's the princess?)
But eventually he and Flynn are BFFs, and they are a fun sort of odd couple. Law&order and the rascal, but they have Rapunzel in common.
Usually Disney horses are helpmates, even if they've got cool personalities, but one of the fun things about Maximus is that he has his own agenda and he's often better at his job than the humans who surround him. (You get the sense that he just puts up with most of them in order to do his job better; does that sounds like any driven sporthorses to you?)
Fun fact: Maximus is also Disney's first - and so far, only - computer-animated horse.