As with chew toys, the wheels should not be made of plastic. Plastic can harm your pet by causing issues with their intestinal area. The pet can chew and ingest the plastic pieces which can cause this. If you use a wheel with spokes, you are asking for trouble. Not only can your pet chinchilla face injury to their arms or legs, the risk of having their limbs amputated increases.
Even your pet can sense when there's danger with these type of wheels. When that happens, they're reluctant to exercise on them. These type of wheels are not recommended to be installed in their cages. If you do use them, you are not looking out for your pet's best interests. Another type of wheel, called a safe wheel, has screws in the inside center. They seem to have more fun on this type of wheel because of the decreased danger.
The recommended measurement for a wheel is 12", even though there is some debate about that. Most people seem to do ok with that size wheel. Pet stores normally sell the wheels with the spokes, which as stated in the previous paragraph is dangerous for chinchillas. There have been reports that the wheel may cause your pet's spine to curve, because there's only so much room; of course, that's probably remain to be seen.
If you find abnormal issues with your pet after getting on the wheel, you should stop until your find out what's going on. If you are unsure about what type of wheel to purchase, contact your local pet store or consult with your local veterinarian.