The first thing to think about when drawing animals is to think about why they're built the way they're built. This informs everything about how to draw them. It influences how they're put together and how much flexibility they have, all of which affects the drawing of them very much.
Animators at Disney frequently talk about the lectures of Dr. Stuart Sumida, who is an expert on animal anatomy. Unfortunately I've never heard Dr. Sumida speak, for the simple reason that story people are never invited to these type of lectures! Only animators (and riggers, who build the characters) seem to be invited to them.
But that's okay, there are a number of tremendous books about animal anatomy that contain the same type of information*.
Anyway, the reason we ended up riding around on horses for transportation is because of how they evolved. They are plant eaters and eating plants requires a lot of intestines to digest. Eating meat takes a lot less in the way of intestines so that's why cows and horses have those big giant bellies and cats and dogs don't. Horses and cows have to carry around all those guts. So in order to carry around all that weight, horses and cows have thick solid spines. That's why we can sit on their backs without snapping their spines!
Also, it's easier on us to ride them because of this reason. Those big heavy thick spines that horses (and similar animals) have don't have a lot of flexibility. When you look at a horse running, the spines stays relatively level and straight. That makes for a much smoother and comfier ride for the passenger.
Whereas animals like cats and dogs don't need to carry around those big piles of guts because it's a lot more efficient to digest meat. Also they have to hunt prey in order to eat. So they developed springier, lighter spines that help them crouch and pounce (in cats) and run fast to take down their prey (both cats and dogs).
If you compare the two running sequences you'll see how much up-and-down movement there is in the cat spine as opposed to the horse spine. Also the cat spine changes shape a lot more dramatically than the horse spine. Cats and dogs have spines that curl and uncurl (or squash and stretch if you prefer) as they run.
So whenever I see a fantasy painting or something like that were someone's riding a jaguar or a lion or something I wonder what that must be like...it must be a bumpy ride!
If you don't have a collection of Muybridge's photos of animal locomotion (both series above are from his work) you ought to get one. Much of what there is to know about animals can be gleaned from looking at how animals move and asking yourself why they move the way they do. This is far more valuable than any book about drawing animals can ever be. That's pretty much how I learned all that I know about animals and I'd say it's served me okay. Looking at an animal and figuring out why they evolved the way they did based on their behavior is a great exercise and there are very few books that bother to do this (if you know of one, let me know).
* Speaking of which, some of my favorite animal books are:
"Animal Anatomy for Artists" by Eliot Goldfinger
"The Artist's Guide to Animal Anatomy" by Gottfried Bammes
"Draw Horses with Sam Savitt" by Sam Savitt