After reading both sides of the debate (cp. beneath), I can't make up my mind!
I live in Toronto, Canada. I buy my mussels from Diana's Seafood or Loblaws.
Evidence for soaking in salt water for 20 mins
Just before cooking, soak mussels or clams in a bowl filled with cold water, salt and about 1/4 cup of flour for about an hour. Why do this? The benefits are twofold: 1. Removing debris [...] 2. Plumping up mussels & clams [...]
3. Using your hands, agitate mussels gently to remove any debris clinging to the shells. Let mussels soak for 15 minutes. During submersion, mussels filter water in and out of their shells as they breathe. Soaking encourages them to expel any sand or debris remaining inside.
Don't fear the mussels - Chicago Tribune by James P. DeWan, Special to the Tribune. DeWan is an instructor at the Cooking and Hospitality Institute of Chicago. Mar 15 2006.
2. Just before cooking, rinse the mussels under cold running water. If they're gritty, scrub them with a brush (photo 1). Discard any with broken or cracked shells. Mussels that feel unusually heavy are probably filled with mud and should be tossed. Tap any whose shells are open with a knife. If the shell closes, the mussel is alive and therefore good. Discard any that won't close.
Now, many cookbooks and chefs alike advocate soaking mussels in tap water for an hour or so before cooking. The idea is that they'll take in the clean water and eject any sand or grit that's hiding inside their shells. Some sources suggest adding flour, cornstarch or cornmeal to the water to encourage the purging. Others suggest these additions will fatten the mussels or whiten their flesh in the process.
However, knowing that mussels are ocean creatures, and me being generally suspicious anyway, I decided to consult three experts: two marine biologists and one "seafood technology specialist." All three agreed that tap water is the mussel's enemy. At best, the mussel will simply shut its shell, precluding any purging; at worst, it will drown.
While one expert said that mussels soaked in fresh sea water (hard to come by here in the Midwest) will purge themselves of some grit, the other two agreed that farm-raised mussels (and commercially available mussels are nearly all farm-raised) are relatively clean to begin with and don't need purging.
My conclusion: Don't soak. Just rinse, and hope for the best.
Store live shellfish, such as oysters and mussels in the shell, in a shallow dish covered with damp towels or moistened paper towels. Never put live shellfish in water or in an airtight container. Scrub shells with a stiff brush just prior to shucking or cooking.
To clean the shells put them under running water and rub against each other for a few minutes. Run under the water until the mussels feel smooth with no visible gritty residue. Never put mussels in standing water (i.e., in a bowl of water): they will open to breathe, releasing the salt water and absorbing the tap water. That's exactly what we want to avoid.