I think the answer to your question depends on the animal that the veterinarian is treating. So, since the animals most commonly seen by veterinarians are dogs, cats, birds, and reptiles, I'll give you a list of some of the diseases most commonly suffered by these animals.
Dog diseases: According to discovery.com, the most commonly diagnosed conditions of both dogs and cats are weight issues, dental problems, and fleas. Dogster.com lists the 5 most common canine physical problems as ear infections, skin allergies, hot spots (pyoderma), gastritis, and enteritis. Dogs can also suffer from urinary tract infections and other conditions. Dogs should also be treated for parvovirus and other parasites.
Cat Diseases: Before adopting a cat, you should have it tested for two deadly diseases: FIV (the feline version of HIV) and feline leukemia. There is no cure for either disease, and they are highly communicable to other cats. Other diseases commonly suffered by cats are urinary tract infections, diabetes, and upper respiratory diseases. Cats can also develop mammory gland cancer, lung cancer, and pancreatic cancer.
Bird Diseases: I'm not as familiar with birds as I am with cats and dogs, so I have to rely on the source for information. According to Alyson Burgess, the 5 most common illnesses encountered in pet birds are Proventricular Dilatation Disease (PPD), also called wasting disease; Psittacosis, or "Parrot Fever"; Psittacine Beak and Feather Disease (PBFD); Polyomavirus; and Candida. I've included a link to Burgess's article so that you can read about these conditions.
Reptiles: Reptiles are most commonly treated for bacterial and fungal infections. See exoticpetvet.net/reptile for more information.
I hope this helps you.