Classes I think would go well with horses:
All Warrior specs - Seems to be generally accepted in the thread already. Knights and all.
Ranger - Horseback archery. Will also give Rangers a much needed boost.
Bard - Gives off that feeling of a travelling songsman. The one who rides into a town, plays a tune for bed and board, then is off the next sunrise.
Classes that MIGHT go well with horses (YMMV):
Ninja - Historically ninja were fairly highly trained and able to use anything as an improvised weapon. Their invisibility and stealth, while partly true, involved a lot of disguise and willingness to dress and act beneath their station (Samurai would barely notice a peasant walking by). Speed is not something they were known for, despite what popular media these days have it as. While they might certainly know how to ride a horse, doing so draws eyes, and that is not what a ninja wants.
Disciple - Being trained martial artists, riding might be an addition to the skills they've learned. Riding a horse would be a means of getting to places quickly, but brings up the idea of a lack of discipline in the disciple. True training might involve actually having to walk to places rather than riding a horse, so this is debatable.
Necromancer - As a Caster who tends to rely on DoTs and help from entities from the Nether (In lore), it might be possible to explain the Necro as a riding class due to not needing to concentrate as much. Maybe. *Shrug*
Classes I don't think should have horses:
Caster/Healer classes - Casting takes time, concentration and focus. These are fairly hard to do while riding on a bouncy animal down a forest trail dodging branches (Extreme example). If there's any distraction while casting a spell, it should fail. While
XenZan puts up a good argument for Mystics being able to use horses (Alignment with nature), it can be argued that a Mystic wouldn't want to force a noble animal into a beast of burden.
Runeblade - I just don't see these guys with horses.
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I also agree that out of combat, every class should be able to have horses, but I'm not sure how to handle the animals simply disappearing when you enter combat. Can't think of another option though. Maybe upon initiating combat, you're thrown off your horse and it bolts? Doesn't quite disappear, just runs off so you can find it and get back on it and hightail it outta there or something.