I wasn't sure so I called the City Council to find out. To my knowledge, these are the rules: dog registration and insurance costs ISK 15,200 in the first year, but ISK 12,000 every year following. Dog owners who take their dogs to obedience school get a 50 percent discount. There are 1,500 dogs in Reykjavík but due to a request by the shopkeepers’ association, they are banned from the main shopping streets downtown. If you live in a building with other people (like on a block with flats or in a shared house), you need the other residents’ permission to have a dog.
The reason for the rules are this: to give the city and its residents legal backing for not being kept awake by barking, and and for not having to deal with dog poop on the streets. This doesn't mean Iceland is dog-unfriendly though. In fact, as a sheep-based rural economy, dogs have traditionally been as important here as anywhere.