It would never happen but I approve of every country making dog and cat breeding illegal for normal citizens. What I mean by normal citizens is just as governments grow illegal drugs for use in government research, government officials would still be able to breed German Shepherds or other breeds for police work, etc., such as bomb sniffing or finding missing persons. But normal citizens would not be allowed to intentionally breed dogs dogs or cats, or let them mate.
What this would do is save millions of animals in kill shelters from being euthanized and it would save millions of animals in no kill shelters from the misery of living in a tiny cage at a shelter, sometimes for years. There would be much more demand for these animals because people would not be able to legally buy a dog or cat from a breeder.
With no humans artificially selecting for specific characteristics, over time it would lead to the end of nearly every dog and cat breed on the following lists. I am perfectly fine with all of these breeds dying out because I'm really sickened by pet breeding and dog and cat breed competitions and all that.
en.m.wikipedia.org
en.m.wikipedia.org
One word about extinction. A law like this would never bring about the extinction of dogs or cats themselves, because there would still be stray and feral dogs and cats mating, as well as irresponsible people who allow their unaltered dogs or cats to roam the streets, as well as the aforementioned government breeding of police and rescue dogs, as well as people who breed dog and cats illegally for the black market dog and cat meat and leather industries.
But it would be easy to prevent black market purebred pet breeding, because vets could be required to report people bringing purebred animals in for treatment for example. And anyone seen walking a purebred dog would be subject to arrest for buying from a breeder, so it would only be a matter of time before pet breeders were found and arrested.
What this would do is save millions of animals in kill shelters from being euthanized and it would save millions of animals in no kill shelters from the misery of living in a tiny cage at a shelter, sometimes for years. There would be much more demand for these animals because people would not be able to legally buy a dog or cat from a breeder.
With no humans artificially selecting for specific characteristics, over time it would lead to the end of nearly every dog and cat breed on the following lists. I am perfectly fine with all of these breeds dying out because I'm really sickened by pet breeding and dog and cat breed competitions and all that.

List of dog breeds - Wikipedia
List of cat breeds - Wikipedia
One word about extinction. A law like this would never bring about the extinction of dogs or cats themselves, because there would still be stray and feral dogs and cats mating, as well as irresponsible people who allow their unaltered dogs or cats to roam the streets, as well as the aforementioned government breeding of police and rescue dogs, as well as people who breed dog and cats illegally for the black market dog and cat meat and leather industries.
But it would be easy to prevent black market purebred pet breeding, because vets could be required to report people bringing purebred animals in for treatment for example. And anyone seen walking a purebred dog would be subject to arrest for buying from a breeder, so it would only be a matter of time before pet breeders were found and arrested.