There is no registry for chickens. Find a show or shows you want to attend, contact the show secretary for entry forms and show pamphlet and enter your birds. Poultry Press lists virtually all sanctioned shows.
One thing you should ask yourself is where your birds came from. If from a hatchery or feed store, chances are that they will not fare well against birds bred for exhibition. If you purchased from a breeder, ask their recommendations on shows in your area.
Some cities allow exceptions for livestock or poultry that are being used in 4H or FFA or other educational programs. You need to know the exact specifics of the code and exceptions for us to give a useful answer.
My cities ordinance says no livestock but the definition of livestock is an animal used for slaughter or profit and my chickens are neither of those. But I'm afraid if I have a rooster someone may complain about the noise. Bit if I can find a way to get passed that it would be awesome