Shrimp can be cooked either shelled or unshelled depending how you will be using them in a recipe. There are various methods to removing the shell. One way is to first pinch off the head and the legs and then, holding the tail, peel the shell off from the body. If shelling frozen shrimp, do not defrost them completely as they will be easier to shell when they are still slightly frozen.
A much-debated question about shrimp involves the need for de-veining. The dark "vein" that runs lengthwise down the back of the shrimp is not actually a vein at all, but the shrimp's digestive tract. Like the other parts of a shrimp, it is both edible and contains nutrients. However, many people dislike the texture of this shrimp part, and they also dislike the idea of eating what amounts to the shrimp's intestine. We've searched for research on the nutrient contents of the shrimp's "vein," as well as the potential contaminant contents, but we have not found helpful information in this regard. Luckily, the vein of the shrimp is easy to see, and if you want to remove it, you can do so fairly easily with a shrimp deveiner. These devices are inexpensive and available at most kitchen supply stores; they make the job of shrimp deveining fairly easy. An alternative method is to very carefully slice down the back of the shrimp with a knife and hold the shrimp under cold running water to allow the force of the water to rinse out the vein contents.