Breed-specific legislation — laws targeting particular breeds — is overwhelmingly a canine phenomenon. Cats are very rarely named in such legislation, and the American Bobtail, a fully domestic breed with a gentle reputation, is not the target of breed-restriction laws anywhere of note.
One area where cat owners should pay attention is hybrid and wild-ancestry regulation. Some jurisdictions restrict cats with recent wild ancestry, such as early-generation Bengals or Savannahs. The American Bobtail is not a hybrid and has no wild ancestry despite its bobcat-like look, so these restrictions do not apply — though owners may occasionally need to clarify that distinction to a landlord or official unfamiliar with the breed.
This information is general and educational, not legal advice. Laws vary by country, state, and municipality, and they change over time.