As a veterinary family practitioner, I most commonly see patients for routine health checkups. That is a good thing, but I am guessing that you are interested in the most common health problems that trigger veterinary visits.
Although I dont have personal quantitative data, I would say that for cats the most common medical problems I see are gastrointestinal upset/diarrhea, dental disease, urinary infections, respiratory infections, skin infections and allergies, kidney disease, thyroid disease and injuries from fighting with other cats.
For dogs (again, this is not quantitative), gastrointestinal upset and diarrhea are the leaders by a mile (thats 1.6 kilometers if you live in the UK). These are followed by dental disease, urinary tract infections, ear infections, limping (usually caused by sprains and strains), skin infections and allergies, and minor trauma to the skin.
For those of you who prefer quantitative data, a major pet insurance company in the United States has published a list of the most common types of claims it receives for dogs and cats. A complete description of the findings can be found here. (Credit goes to Veterinary Pet Insurance, Inc.)
To summarize their findings, the top ten causes of claims for each species are listed below in order from one to ten.
Dogs: Ear infections, skin allergies, hot spots, stomach upset, intestinal inflammation/diarrhea, bladder disease, eye infections, arthritis, thyroid disease and sprains.
Cats: Urinary tract infections/disease, stomach upset, kidney disease, intestinal inflammation/diarrhea, skin allergies, diabetes, constipation, ear infections, respiratory infections and thyroid disease.
Here are my tips for preventing these problems. Feed a high quality diet. Practice home dental care (brush your pets teeth). Keep your pet free of parasites such as fleas and worms. Maintain your pet at a healthy weight. Do not allow your cat to go outside. And finally, take your pet to the vet for regular checkups to detect developing problems when they are at an early, treatable stage.