It's either a complicated answer, or a really simple one, depending on how you look at it.
Busking laws usually vary by city. Some cities (like Fresno) don't even have laws regarding busking so it comes down to the individual police officer whether or not to bother you. Usually if you just say sorry and that you'll pack up and leave right away you won't have a problem. Because of this most buskers use a "Better to ask forgiveness than permission" approach. Meaning they don't worry about it, just set up the show and start performing. If you get shut down, you get shut down.
Otherwise Krab is right - The town hall will be where you get a permit if there is one. Again this will vary by city. Some cities you have to audition and pay a significant fee, other cities you don't. As I said previously, Fresno doesn't even have busking laws. Providence requires a street performance license, but getting one just requires two trips to City Hall and a $1 fee. I believe in Boston you are restricted to certain areas without a license. Something like that, it's been some time since I looked at Boston.
I know in the US busking has been declared a legal right for the most part - the argument being that you can walk into a public space and perform, you can walk into a public space and beg for money, and you can walk into a public space and talk to people - there's no reason you can't do all three at once.
Which is all a long way of saying: You'll either have to risk it, or go to City Hall.
