Showing at the county fair level, just make sure they are clean and healthy looking. They pick the prettiest birds usually, unless there are breed judges in your area. I put in my Indian Fantails and a couple grizzles and ended up with two first places. My fantails are not perfect by the standard, but they are very pretty. And my homers were off my flying team so they were in good shape, and the nice colors helped a ton. County fairs around here don't appear to be too keen on the chicken breed standards either, but the ribbon winners did look pretty good. Nothing compared to a real pigeon or chicken show, where there is a qualified judge for every breed.
As far as checking for things wrong with the birds when entering them....my pigeons didn't have to be checked for anything, but my chickens were banded with those metal crimp-on bands and I think they took a blood sample or something to make sure they didn't have a serious virus.