Short answer
Mussels, clams, and oysters are all similarly safe for dogs when prepared correctly. The key is that they must be thoroughly cooked (never raw, due to bacteria) and completely removed from the shell to prevent choking or gut injury. All shellfish should only be given in very small amounts as an occasional treat. There's no significant safety difference between the types; the risk lies in the preparation. Offer a tiny piece of plain, cooked mussel meat and watch for any stomach upset.
Mussels — Food safety facts
Safe to share · Dogs · full breakdown on the linked food page
Cooked and removed from the shell, in small amounts. Raw shellfish carries bacteria.
The full verdict, portion guidance and serving notes are collected on the Mussels food facts page.