I know that at the shelter I volunteer at, cats are tested for FIV and FeLV, wormed, get flea/mite treatment, and are given a general wellness exam, brought up-to-date on shots, and have a stay in isolation before being made available for adoption, and a vet comes in once a week to look at any animals that appear to have developed a health issue during their stay at the shelter. However, upper respiratory infections and herpesvirus outbreaks are pretty common and widespread in shelters because of overcrowding and stress (there is just no avoiding them), and minor viral infections can turn into major bacterial infections if not monitored carefully. For that reason, it's very important to be careful when introducing a shelter kitty into your home, and to get the new cat to the vet for a full exam as soon as possible. And, many shelter kitties are of unknown origin, and may develop health issues that just are not apparent while they're at the shelter. The shelter workers try their best to monitor the animals for signs of disease, but it is no substitute for the care of a loving owner who knows their cat. It's a big financial responsibility to take in a pet, and we usually tell people to have the kitty vet-checked as soon as possible, and to be financially prepared to start the new kitty and resident kitties on preventative antibiotic treatment if the new kitty proves to have a URI.
Best bet is to ask the individual shelters what provision they make for health/vet checks on the adoptable cats, what kind of things they test for, what immunizations the animal has had during its shelter stay, and what health issues they believe to be of most concern for their cats being adopted.