Vets aren't normally experts in chinchillas. Neither am I. This list of common health problems is just my advice. Use good judgement in assessing your chinchillas problem. Remember many issues can be resolved by reducing stress. Also if at all possible, avoid surgery. Even with an experienced vet, your chinchilla has a 50/50 chance of survival every time it goes under anesthesia.
Ringworm/ Fungus:
Symptoms: Fur loss.
Treatment: Put antifungal cream on the infected area. Only buy a prescription from your vet if you are sure the product is not sold over-the-counter in the store.
Kink in tail:
Symptoms: Tail kinked, less motion in tail
Treatment: Pinch the end of your chinchilla's tail. Your chinnie should react at least slightly. The end of the tail should also be warm. If it is not either of these things, take your chin to the vet. Also make sure your chin is not dragging its tail or stepping on it. Take your chin to the vet if you see any of these symptoms.
Diarrhea:
Symptoms: Wet stools, squishy/messy stools
Treatment: Feed chin dry toast, reduce pellets. If the chin has been taking antibiotics, feed your chin bene-bac to improve their stools. Only feed a quarter of a tube or the amount directed on the package.
Fur biting:
Symptoms: Chin pulls out its own fur.
Treatment: Reduce stress by separating him/her from other chins and put it in a dark, quiet room.