I selected my dog (and soon another dog) for their characteristics. I wanted a small active dog that really had a character, so I picked a Jack Russel. With a crossbreed, to quote Forrest Gump, life is like a box of chocolates. You just don't know what you get. Also from my beliefs, pure breds are healthier than crossbreeds, but I guess that just depends on the breed of dog.
A lot of people select a dog on the way it looks, which to my belief is a mistake, but because of this, dogs like the Labradoodle are getting more popular. They just look better (according to some). I'm not entirely sure if here in the UK, purebreds are more popular then mutts. You see a lot of rescued dogs from southern Europe where they lived on the streets and thus over the years have mixed with God knows what.
Also, if I may correct you, the pitbull isn't a breed of dog. It's often used to mention the AmStaf or the English Staffordshire Terrier. Sometimes even the bulldog and the bull terrier are called pitbulls. But it's not a breed :)