Here's how I get to work at home when I bought fresh mussels.
1. Put the mussels in a large bowl or pan, sprinkle with a good pinch of salt and add tap water until the mussels are fully submerged. If you aren't going to cook them right away, put the mussels in the fridge. Remove the mussels from the fridge one hour before you intend to cook them. Let them rest at room temperature. You'll see that in the end there will be a fine layer of sand on the bottom of the bowl or pan.
2. Here's why. Fresh mussels are (by lack of another word) comatose because they've been kept in refrigerators for a couple of hours before they finally made it into your kitchen. And the mussels are dehydrated because they are not kept in water. So adding them back to water (especially salty water) will revive them. Give the mussels some time to wake up. They will start spitting out sand that is still in their shells, which is a normal cleaning process.
3. All this is to our advantage. First of all it'll be much easier for you to spot a dead mussel. So if an open mussel that has soaked for a while in water still doesn't close when you press both shells shut, it's probably dead. If you're not sure, check the smell. If the mussel smells bad, you definitely have to get rid of it. Also remove all the mussels with broken shells. Secondly, the mussels will be less gritty if they've had time to spit out sand.
4. Strain the mussels and discard the salty sandy water. Rinse the mussels well. With the back of a knife blade remove any barnacles or other impurities on the shells. It should all come off easily. This goes best under running tap water. Also remove any hairy bits coming from inside. Just rip them off. They are inedible and unpleasantly chewy.
5. Once the mussels are all cleaned, transfer them to a large sieve and then rinse them again. The mussels are ready to be cooked now!