Can you talk to your neighbours and see if their would be a problem? I'm out in the country, (and I don't have goats yet), but I would think if you got the ok from your neighbours, that would be a good place to start. If you get two or more, pluse give them lotsa toys,, they shouldn't make that much noise, and personally, they sound nicer thatn a barking dog, unless they are CRYING. If you are home all day, you could also be there to check on them when they do start. good luck!
I wouldn't risk it. What type of fines would you get? How many neighbors would you alienate? Better to rent a stall, yard, pasture close by.
If your children want a goat and are in 4H, or if they are old enough to join (5 - 18 years old), you might ask if there is a 4H ranch in your area. If there is, you child might be able to keep their animal there.
Another alternative might be to buy a share in a goat.
I agree about not risking keeping goats illegally. Most goats are so lovable and you and your family will most likely get very attached. You wouldn't want to go through the heartbeak of having to rehome a member of your family (the goat LOL).