Greatist has a complete article about buying shrimp. I found it interesting in that most shrimp sold at a seafood markets have been previously frozen. They spoil very quickly and that is also why they don't usually sell shrimp with the heads on them since that lessens the fresh quality of the shrimp. Shrimp should have a firm texture and not an "off" fishy odor. So unless you buy them freshly caught, it is best to buy shrimp frozen.