I work at a dog boarding place in an affluent area. There are purebreds, there are mutts. The stigma has almost nothing to do with purebreds themselves - it's the images of the owner that are evoked.
Purebred owners in my area overwhelmingly tend to:
- Believe the cost of their dog makes it more valuable than others
- Believe their dog's breed characteristics make basic training unnecessary (golden retrievers have a pretty high bite rate, probably due in part to this)
- Have dogs as a statement piece, a well-groomed pup who is part of the family image, but is not taken care of as the individual being it is. It's the dog who lays on his dog bed bored all day, and has access to the backyard, and the owners walk by every few hours and pet and manhandle the dog, and think they're giving the dog everything it could ever want.
- Keep their dog at home rather than exposing it to new places, dogs, or people. When I'm out and about, it's the mutts who get to come along and go places. When I run through the suburban neighborhoods, it's the purebreds who are losing their minds barking from the window or the yard, bored to death.
On top of that, there is a popular opinion that it makes no sense to buy a dog from a breeder. I have to admit, breeders are not solving a dog-demand problem, so I question their utility overall. They seem to mainly be standard-bearers for the american notion of what their dog breed should be.
To boil it down, the stigma is this: buying a purebred is a calculated decision made by wealthy folks in their own interest, and adopting a mutt is an act of love done in the interest of the dog. It's not that clear-cut, but there's the stigma.