Actually, live is how it should be done with all shellfish, save for scallops which are cut free of their shells beforehand.
Every time I get an order of mussels or clams, I check for dead ones and dispose of them before cooking.
As for clams, I will feed them before they hit the water. Placing your clams in water with corn meal causes them to eject most if not all of the sand they are carrying in exchange for it.
The best thing though is to kill them instantly - take your shellfish and place them in the freezer for awhile, and they will go into a state nearing hibernation. The moment they hit boiling water (or in my case hot steam - I have a big steamer pot for shellfish) they are killed. Doing this really helps with lobster, as they are killed so fast they don't have time to curl up into a ball.