Dogs shouldn't be allowed in public, owners should only be able to exercise them on their own private property. If the dog needs to be transferred from one property to another it could be secured in a car or van for the journey.
There'd be exceptions obviously for things like police dogs and possibly guide dogs, but zero tolerance for dogs kept as pets.
I'd like to see dog laws to be somewhere between knife laws and gun laws. Like a knife, a dog should only be allowed in a public place where there is a legitimate reason to be in possession of one. Just as "feeling safer" is not a legal defence for carrying a knife, "taking it for a walk" shouldn't be a legal defence for having a dog in a public place.
The comparison to gun laws comes in with the idea that, like when someone wants to own a gun, first they should have to undergo tests to ensure that they are a suitable person to own a dog and that they have the facilities to ensure it is kept safely on private property away from the public.