I suspect that we are on the path to dog licences. If it was the case that you had to register your dog's microchip, prove that you had third party insurance and possibly take a basic test of husbandry, I'd be fine with it, so long as it was actually enforced and the money used for the local warden services, rather than being just a tax on the sort of responsible dog owners who would sign up.
Because enforcement is one of the issues. Responsible owners end up carrying the can for the feckless, ignorant and arrogant. Currently, dogs are allowed off-lead on our local playing fields, but I can see that changing in the next few years. Not because of the majority who pick up after their dogs and keep them under reasonable control, but because of the minority who let their dogs shit everywhere and have zero recall.
The other major issue is people who get a dog that they are just not equipped to handle. With some dogs, you can get away with this. Other breeds are not so amenable. They have dog fighting genetics, or guarding instincts bred into them, or they require vast amounts of exercise and stimulation and become frustrated and wild if they don't get it. Possibly we could do as they do in some parts of Germany (IIRC), where if you won certain breeds you have to accept additional restrictions - we could require someone who wants a malinois or an XL Bully to pass a test and take the dog to training classes.