Some (predominantly woodland or mountainous) goblins ride wolves/worgs. Some (predominantly plains, hills, or swamps) ride a homebrewed long-legged scaly bird (a dinosaur, really, I suppose, these days. Since now we think lots of them had feathers) -think a reptilean-ish axe beak or feathery velociraptor, with bird head/beak, not dinosaur snout/teeth/mouth), called a Zarx. Uncommon but not unheard of, cave-dwelling goblins have been known to train/befriend giant bats to use for air transport.
My setting's analogue of Hobgoblins tend to larger "troops transports" than individual mounts. So, giant lizards or fiendish elephants, fitted with a enclosed howdahs. Armored carts or large chariots pulled by teams of bonnacon ("gorgon'esque" creatures. I expect they were originally the medeival lore of a gnu/wildebeest. But I've turned them into an actual fantastic creature in my setting. So, think "an ugly antelope with curved horns and scaled/armored hide with acidic gas breath weapon"). High-ranking officers/commanders might ride a bonnacon, themselves. The "hobgoblin" nation's upper eschalons have wyverns for aerial forces.
Dwarves are prone to bears in the hills/low mountains. Big-horn sheep/ibex-like caprines in the high mountains. In colder climes, woolly rhino or mammoths might be used for transporting large numbers of troops but are more likely to be beasts of burden to haul seige weaponry, portable smithies, armaments, supplies, and such.
Elves ride stags, of course. Elk in cooler climbs. High-ranking officers or nobility might have megaloceros ("irish deer," what Thranduil rode) where they can get them. There are squadrons that ride giant intelligent/speaking hawks. Royal houses might breed/raise/train griffons. Some, very very rarely, are aligned by unicorn or [homebrew] "Silverfawn" - white/silver elk, sacred to the silver moon goddess, who can teleport (folklorically, only under/in moonlight, but that is not so).
Orcs tend not to use mounts. They are predominantly mountain and largely subterranean dwelling. So going by foot is more practical. And they move/climb better and faster through such terrain than most bipeds (their quarry) can. Beasts of burden will tend to giant boar (above ground) or lizards (below). So commanders might be mounted on one of those or some other unique monster/creature.
I have the remains of a fallen human empire, now exiled from the mainland. But legends about them say they rode dragons (really, just winged drakes or, think sting-less wyverns). But no one has seen a Selurian in a couple of centuries now and they exist more as villainous bogeymen in stories to scare children.
I have a race of demon-worshipping mutated humans, now reptilean/snake men. (another/different, and much older than Seluria, fallen empire) Those who do not possess serpentine bodies, will use giant basilisk, dracolisk, giant snakes, and all manner of chimeric creatures that they augment/create with their wicked alien magics...if when anyone is unfortunate enough to cross paths with a force large enough to use mounts...which, probably, no one would hear about anyway, with no survivors.
I don't think of bugbears as generally needing or wanting mounts. My world's analogue of gnolls wouldn't either. The alleged "Dark elves" are so far lost to legend and myth, who one knows what they'd use...they're not even real anyway, right?
That's really about it/all that's been developed in this area for the setting. And, also, humans, elves, dwarves, and goblins, hobgoblins, orcs, (and the Gorgon snake-demon-men if/when they make a play to re-conquer the world) are the only species/societies prone to assemble large "wartime" forces.
EDIT: To
@aco175 's point, the human realms -and those traveling within them, so as not to draw undue attention- are the ones most prone to use horses for battle/war mounts.
Halflings use ponies to travel or work animals...but they aren't engaging in "war." Cows or ox, even a good sized ram or goat, are better for plow-pulling than a pony. Large dogs/mastiffs are most likely to be used as a battle mount...but that's really more for those wilderness roving weirdos. /EDIT
