Refrigerate live shellfish in a dry dish covered with moist seaweed (if you have access to it) or damp paper towels; do not store on ice, in water, or in plastic. Because oysters naturally open and close their shells, they need to be stored cupped-shell down so that their liquor doesn’t drain out. Correctly refrigerated, clams and mussels will last up to two days and oysters somewhat longer, up to one week. Cleaned squid should be stored in leak-proof bags.
Cooked shellfish should be stored separately from raw to protect against any cross-contamination of bacteria from the raw shellfish.