I think the muslin I bought for my backgrounds, one white, and one dyed black, are too thin. Are the muslin backdrops online made out of thicker material than that which you buy at the fabric store? The white one seems to show through too much. The black is ok.

Can anyone help me out here?

They should be approximately the same thickness. The white will be alright with front lighting, especially a background light, but if you want to take care of the "thinness" it's simple enough to paint one, in this case white. It will be stiffer, but that's the case with all painted muslins. We have several, both commercial and those we painted ourselves. Rather than folding them, we push them into large stuff sacks or duffle bags for storage.

There will be some wrinkling, but it relaxes after it's hung awhile. If space permits, you can also arrange good separation between the backdrop and your subject, then restrict your DOF in order to conceal the wrinkles. When sufficient space is not available in location shoots, we use a gausian blur layer in Photoshop to blur the backdrop.

Basically thin is good, because it helps in draping muslins and is one of the big pluses in using them rather than rolled backdrops. We have 14 rolled backdrops hanging in our studio plus features for papers, but still find uses for the muslins there whether to cover a larger area for large groups or to continue the drape onto the floor under subjects. Our biggest is 12x20, but on a couple of occassions we have had to mate it with an identical 12x12 muslin to get enough expanse for the job. And that's the key for muslins: Versatility.