As healthy as chinchillas are, there are still some issues to be aware of.

1. Teeth issues. A healthy chinchillas teeth should be a yellowish/orange colour. Should you see white teeth, chances are you have a calcium deficiency in your chinchilla. Another tooth problem is malocclusion. Two great articles about it can be found here www.paw-talk.net and here www.pubmedcentral.nih.gov Symptoms of malocclusion include: drooling, eye problems, lack of appetite, unexplained weight loss and pawing at the mouth.
2. Fungus. Also known as ringworm, this can be a devastating disease. Noticed by bald patches on the skin, it is highly contagious and will spread to all the animals in your household (including yourself) if you are not careful. If you notice bald patches on your chinchilla, one of the easier ways to determine is it is fungus is to acquire a black light. The fungus will illuminate under the light. Also, the fur at the edges of the bald patch will pull out easily. Upon discovering fungus, remove and quarantine the chinchilla from all other pets. Be sure when handling the chinchilla you that you do not touch any areas of your body and that you wash your hands well afterwards. Thoroughly dry your hands as well as the moisture will keep the fungus alive. For treating the fungus many have used Tinactin (available at your local pharmacy) with success. Purchase the powdered kind and mix into the dust bath. Use 1 TBS per bath. Keep the chinchilla quarantined for 8 WEEKS minimum. It is also a good ides to do a treatment for any chinchillas they may have been in contact with the affected chinchilla to prevent an out break A good article can be found here www.etc-etc.com/fungus.htm.
3. Hair rings. Affecting only males, this is when the fur and debris get wrapped around the chinchillas penis causing a tourniquet effect. Symptoms include excessive cleaning, red, swollen penis/protruding penis. It is advised that monthly checks are done as a proactive preventative. Should a hair ring be found it needs to be immediately removed. With frequent checks most are easily removed at home. You can use your fingers or a pair of blunt tweezers to remove the ring. To extract the penis from its sheath pull the skin of the sheath back, then hold the tip of the penis firmly and pull it all the way out. Be sure to fully extract the penis from it's sheath as hair rings can be found along the entire length. This task it usually best accomplished by two people. A quick video on hair ring removal can be found here www.spoiledchins.net/hairring.htm. Should the ring be wrapped so tightly you can not remove it successfully at home, seek vet attention IMMEDIATELY.
4. Soft stools/diarrhea. Many things can cause this in chinchillas. Change of environment, stress, symptom of another illness & internal parasites. Upon noticing soft stools, remove the pellets from your chinchillas diet and feed large quantities of quality hay and have plenty of fresh water available. Be sure your chinchilla is drinking well as soft stools will dehydrate your chinchilla. You can also give a piece of shredded wheat or blackened toast. Should the symptoms persist for more then a few days, seek a vet a get a fecal smear done.
5. Fur Chewing. While not a medical condition, fur chewing is something that crops up now and then. Commonly found with poorly bred or stressed chins. Best preventative is to buy quality animals and keep the animal in a stress free environment. For many chinchillas affected with fur chewing there is no real cure. Some start to chew when a mate dies, others chew from birth, still others from simple environment changes. If you can, find out what has changed and reverse it. If the chinchilla lost a life long cage buddy, you can try a stuffed animal to keep it company. Be sure if they start to chew it that it is removed!
6. Heart mummers. Heart mummers have been on the rise lately, especially on the west coast of the US. It is a problem that has gone unnoticed for the most part and a lot of breeders do not test for this defect. Heart mummers or other defects can usually be discovered by a vet trip. Should a mummer be found, the vet can recommend any needed course of treatment. Most chinchillas can live long lives with this defect, however should not be used in any breeding program. To help screen any chinchillas you plan to purchase you can ask your breeder to provide a health certificate to prove the chinchilla has be screened. Most breeders will do this at a charge to the buyer.

For some great picture references, look here: www.sunsetchinchillas.com/Pictures.shtml.