If you look at the different breeds of dog you will see a lot of variety both in shape and size. Just compare a chihuahua and a great dane.

However, if you compare different breeds of cats they are less varied than dogs (at least in my opinion).

Assuming that dog and cat breeds are a result of artificial selection using one initial 'type' (wolves in the case of dogs) then why is it that we can have such a large variety of phenotypes in dogs but not in cats?

Good question and this has in part been discussed before on the site. See
http://www.askabiologist.org.uk/answers … hp?id=7416

As to why polymorphisms in the IGF-1 gene in dogs produce such a large variety in size and not in cats is, as far as I am aware, unknown.

One point to note is that domestic cats are  - and probably have always been -  kept principally as companion animals.  Although they may "earn" their keep with some  pest control duties they haven't been under anything the type of hard artificial selection that has been practised on dogs.

As well as providing companionship dogs are used for hunting (anything from deer to rats), herding (sheep, cattle), guarding, fighting, pulling sleds, hauling nets etc etc,  and we have selected for breeds that are fit for the job we wnat them to do (in the climate/environment we want them to do it).

An explanation that I heard as a graduate student was that the skull of a dog tends to undergo much greater change in shape than the skull of a cat as the animal matures. In particular, the snout tends to be short in a newborn puppy and proportionally much longer in an adult dog. Growth-related effects like this are often subject to a lot of evolutionary variation; a dog in which snout elongation starts early in life, proceeds quickly and ends late will end up with a much longer snout than a dog in which elongation starts later, happens more slowly and/or ends earlier, so there are a lot of variables in play.

In other words, the fact that skull shape changes during maturation leads to variation that breeders can exploit by artificially selecting long-snouted or short-snouted individuals. With cats, this type of variation in skull shape isn't available.

I don't know whether other aspects of body shape, such as the lengths of the limbs relative to the body, also change more during growth in dogs than in cats. However, the point about skull elongation is probably a partial explanation for the difference in breed diversity mentioned in the question.