From the Icelandic sagas we know that people in the Viking period, peasant and king alike, had dogs as pets, mostly for guarding property and as hunting companions. We also find dog bones as burials, sometimes buried as if it was a person (without gravegoods, but buried with care and in 'sleeping' position), whereas horses were more commonly killed violently or decapitated, or sometimes dismembered, suggesting a more sacrificial nature of the animal rather than as household companion (or even another nonhuman 'person'). However, the borders between a valued working animal and a pet are a bit blurry sometimes (just like today), like how in the Ladby ship burial there were four dogs and eleven horses accompanying the deceased, and in Oseberg four dogs and a number of other farm animals as well. Both horses and dogs were part of the regular sacrifices at Uppsala and Lejre if we can trust the later Christian sources. Dogs would already come in a wide variety of breeds and sizes, from small terrier-size to shepherd.

Cats were also known, though particularly as the animals of the goddess Freyja. I do not know of any archaeological finds of cats. Around this time the Roman/Celtic-introduced cat would have been replacing the native ferret as a farmyard pest control animal.

Interestingly enough, the Icelandic Vikings also had bears as pets, which was made illegal by the popular assembly. Having a brown bear was most common, but as Scandinavians went out across the North Atlantic, they also tamed wild polar bears to sell as pets to European royalty.

This page has lots of interesting info on Viking pets and live stock.

Some cool trivia:

There was a tradition that a newly married couple had to get a kitten before getting their first house since a house is not a home without a cat. These cats would have been mostly Norwegian forest cats which are a wild breed native to Scandinavia, although nowadays they have interbred significantly with domestic cats. They are still a common and popular pet breed.

Because of their large trading networks the Vikings had pets from as far afield as India, such as Peacocks owned by the elite.

The first Icelandic bishop gifted a polar bear to the German Emperor.