Chihuahuas get hydrocephalus. Great Danes and St. Bernards bloat and get twisted stomachs. Springer spaniels tend to fracture their elbows, while labs, especially if they love to play Frisbee, tear ligaments in their knees. German Shepherds are prone to hip dysplasia. Dachshunds and Basset Hounds get back problems and the leg bones in the tiny little toy breeds are so thin that repairing fractures can be more expensive, and very difficult.
Cats have their problems too.
White cats, with pink skin, are prone to sunburned ears, which can lead to skin cancer. White cats with blue eyes are usually deaf, but then you wouldn't want a black cat. They bring bad luck. And if you want to consider your poor veterinarian and his or her overworked staff, don't get a really big dog. They are too hard to lift up on the exam table.