Things to know about your pet chinchilla
Fur Fungus
This is the most common problem in the area of health concerning chins.  A form of ring worm, be aware that this can be transferred from animal to animal and sometimes to humans. 
Keeping everthing clean will help prevent the problem, as it is much easier to prevent then cure this ailment.
Stress can also trigger fur fungus as well as dampness and humidity.
Fur fungus can be identified as a scabby/scaley patch of skin often seen on the head, face or front paws, but can happen anywhere on the chinchilla.
Treatment can be simply as adding an  antifugal powder to the dust bath for a few weeks.  This could be a foot powder (Desenex or Tinactin) or other (like Vagisil).  Thorough cleaning of all equipment and cage  is in order. If you do not clean the cage, the room and area, along with all the equipment completely and treat the animals, you will find there is a vicious circle of reinfection.

For chronic or repeated infestations a talk with your vet is in order.
FUR CHEWING or Fur Biting
A Y!A question
Fur chewing is said to be genetic, and triggered by stress.  This was a devestating problem for early ranchers (pelts and quality breeding animals were very highly prized and expensive), and the ranchers would cull and destroy the animals and even the entire lines that carried this problem.
Pet owners and hobbyists are less severe and can not imagine killing an animal that has this problem.  But this is one issue we have seen more and more with the advent of more hobby breeders.

ANY CHIN THAT IS KNOWN TO BE A FUR CHEWER should not be retained in any breeding program, and all related chins should be suspect.

Some female chins duing lacation will overgroom either her own tail hair or that of her young or mate.  This is not considered fur chewing by most breeders, and ceases once the litter weans.
Shedding and Fur Slipping
Chins can shed almost continously, though spring and fall are the heaviest times of noticeable hair loss.  Regular
grooming and dusting will help keep this to a minimal problem.
FUR SLIP, a condition that is believed to have evolved in the chin as a defense to predators.  When nervous and grapped, the chin will lose a large portion of hair that literally slips from the coat, leaving the animal that was going to make it a meal with a mouth full of fur, or a keeper with a handful of fuzz!   Highly nervous conditions and improper air flow and enhance the chance of FUR SLIPPING.
Some say that a chinchilla will drop off its own tail to escape a predator, but this is a myth.  Pet chins do not stay very tame if they are picked up by the base of the tail, and a chin can have its tail tip actually pulled off if the chin is not handled properly.  But popping and dropping the tail is a trick known to lizards called Skinks.