White Shrimp
Louisiana shrimp are the world's gold standard for shrimp, and white shrimp are the better of the two major species we produce. Although not as many white shrimp are caught as brown shrimp, the white shrimp crop are worth more because they tend to be bigger, and the bigger the shrimp, the better they taste, and the more they're worth per pound.
White shrimp migrate from the Gulf into the estuaries along the coast in the late spring and early summer, and stay there, fattening up, until things start to cool off in the fall. Fall is the white shrimp season, although some turn up in the spring in some years. Although to the untrained eye white shrimp and brown shrimp look pretty much alike, white shrimp are the ones with the ridiculously long antennae.
Shrimp are sized according to "count"--the number of shrimp per pound. The best for barbecuing or broiling are under 20 count. Most of the shrimp you get in more elegant dishes are 20-25 count; for things like fried shrimp or shrimp Creole, they go as small as 40 count.
There are no better shrimp for barbecue shrimp, grilled shrimp, broiled shrimp, or any other dish where big shrimp are needed. The meat is a bit tenderer than that of brown shrimp, and the shells aren't as hard. They're easier to peel, it seems to me. And at least one authority notes that white shrimp don't eat acorn works, which is what gives some shrimp that pronounced iodine flavor at certain times.
Although getting them fresh off the boat is clearly the way to go, white shrimp freeze and thaw without any significant damage to texture or flavor. So they're available year-round, not just during the fall shrimp season.
Worst shrimp dish, if you ask me: fried. Best: New Orleans-style barbecue shrimp, as created by Pascal's Manale.