The beaches are probably cleaner in Edisto or Isle of Palms - MB is very touristy. North Mrytle is cleaner than Myrtle. But since the season is pretty much reduced now that school has started, it won't be as bad.
Here's the website for the list. There's alot of info on it:
Dog Beaches
I forgot to mention, the cleanest beaches are in the residental areas. Away from the hotels. If you want to have some fun, go to the end of Cherry Grove(N. Myrtle). Go down to the beach and walk around the tip to the backside where the inlet is(only a few hundred feet). You have to do this during low tide. You can see bunches of hermit crabs changing shells in the tidal pools. You just have to be careful if the tide is coming in - you can get trapped and it can get scarey. If you want to hunt for sharks teeth, go behind the Gay Dolphin in Myrtle. Or just walk the beach and look for areas with lots of crushed shells. Once you find your first one, they are easy to spot. Bird Island is known for finding really great shells. There's tons of things to do there. Contact the Chamber of Commerce for some info. They'll send you books.