Since I’ve already covered the avian steeds of the Final Fantasy world(s), I think doing the same for the Mushroom World wouldn’t be a bad idea. There’s not quite as much to say here, though. Super Mario Bros. 2 introduces ostriches known as Ostros, which are ridden by Shyguys.
You can ride one yourself if you subdue the Shyguy (or, for that matter, just stand on top of it). Also, if the Princess stands on an Ostro’s head, it totally looks up her dress. Anyway, Ostros only appear in this one game, and the credits and instruction booklet misidentify the Ostro as “Birdo,” and vice versa.
An ostrich IS a bird, of course, but it’s obvious that the Ostro was meant to be the ostrich character. Later games confirm this by making it clear that the egg-spitting animal is indeed Birdo.
For more information on Birdo, try this post by Drew Mackie.
Ostros appeared occasionally on the Super Mario Bros. Super Show cartoons, where they were largely used as beasts of burden in place of horses.
If I recall correctly, they showed up quite frequently in episodes with Wild West themes. There actually were a few episodes that had horses, but not that many. I don’t know of any appearances of horses in the actual games either, although apparently there are a few in at least one of the Mario Party titles. In Super Mario World, Yoshi becomes Mario’s main steed, but Yoshis probably wouldn’t allow just anyone to ride them.
I’m not sure why we haven’t seen any more Ostros used for riding or cart-pulling, but I’d like to. Then again, I say that about pretty much EVERY creature to appear in a Mario game, don’t I?