For owning? Mutts. I know they're not a breed, but they're a category, at least.

Just in general? Borzois. They are absolutely, utterly drop-dead gorgeous, and elegant, and sweet. They're also utterly useless, rather dumb, and fairly obscure and uncommon, but they're just so pretty... they're the sort of dog you get because they accessorize well with your velvet chaise lounge, not because you want a pet to romp and play with.