I think neutering for pet dogs( not necessarily working or show) along with microchipping should be made law. To reduce numbers of dogs and make people have to pay something for them, it might just make some people think twice about keeping them. Not many, but even if a few people didn't it might help keep the population down.
So how do you decide which are which?

The majority of show people over the years have started off with a so called 'pet' dog that someone said oh that's a nice dog, why don't you show it'
If your neutering scheme had been in place 40 years ago I, amongst the thousands of people I know in dogs, certainly wouldn't now be enjoying a wonderful, fruitful, enjoyable, friend making hobby. I didn't know when I bought my first Cocker that she would win a qualification to be shown at Crufts. Nor did I ever think I would go on to import a French Breed into Scotland and work them.
I don't know what the answer is, but it certainly isn't to neuter everything. I certainly agree that there should be better control on numbers - I know of one woman in England who has a Brittany dog and numerous Springer bitches and is breeding them to produce 'Bringers' on the bitches every season. She has been reported to various bodies who have believed what she has told them without checking the internet sites she sells the pups on.
My answer would be to bring the Kennel Club's Assured breeder scheme into Law, and that anyone who wishes to breed must adhere to the strict regulations for the rest of us - numbers of litters from a bitch, minimum and maximum whelping ages, health tests, after sales care, taking back dogs that require re-homing etc. That would cut numbers straight away. Any matings happening outwith that scheme would have to be stopped by the mis-alliance injection, and the bitch then spayed. If it got past that stage the pups would be taken into the care of the authorities when weaned and neutered when old enough. The only problem is it would need funding - and no-one wants a dog licence in case the government relocates the money instead of ring-fencing it.