Of course he has. It's called Superdog. Born on Krypton and to his parents Supermum and Superdad. Loves Hot Dogs and Hot Cats. Look out for him in the latest super comics.
According to the DC Comics history of Superman, when he was a boy growing up in Smallville, Clark (Kal-El) did have a dog, another survivor of his home planet, who he named Krypto. As far as I know, Krypto did not accompany Clark to Metropolis and may have died before Clark was fully grown.
Superman's dog is named "Krypto". His dog is a member of the Legion of Super-Pets. Krypto wears a yellow dog collar and a flowing red-and-yellow cape modeled after Superman's.