Erm. I think Elves would have hunting dogs, most likely: maybe not big packs of pursuers such as beagles, but certainly retrievers, pointers and birding dogs! They had Huan, after all, that we know of, and he was a hunting hound. Hunting dogs make sense with Elven choice of weaponry and lifestyle, also.
I feel like Hobbits would go in for the big packs; the more the merrier. Farmer Maggot had his dogs, Grip, Fang and Wolf, who were intimidating dogs that he would threaten to sic on people found stealing his mushrooms. But the dogs were actually big softies who wouldn’t harm anybody. They were trained only to scare thieves, not hurt them. They were so effective at this, in fact, that Frodo was still afraid of them thirty years after his youthful mushroom-rustling misadventures 🙂
I feel like sniffers/truffle hunters would also be highly prized by Hobbits. Because FOOD. Also sheepdogs. They would LOVE smart, intuitive, sheepdogs. Oh. So. Much.
idk about Dwarves! Perhaps burrowers or ratters? I feel like they would admire the go-get-em spirit of terriers. Also, a lot of Dwarf settlements seem to end up by rivers/the sea: perhaps they like water-dogs and fishing dogs such as spaniels! Which would be sorta awesome.
I wonder if it was the ancient Numenoreans who bred the fancy show dogs. It feels like the sort of thing the Numenoreans might do. Their mutt descendants might still be sniffing around Gondor, the carefully-purebred traits and characteristics all but erased.
Aw Yiss 