You can ride an ATV on any enrolled street, which may include some county roads (Fremont and Carbon counties have some enrolled roads). It is up to the rider to determine if a road meets the requirements for ATV riding. In addition, you may ride on roads that have not been enrolled in the program provided your ATV is street legal, meaning you have a motorcycle license plate and proof of liability insurance.
Check out these Granger, WY ATV/OHV resources:
