Southern Californians are blessed all year round with coastal asparagus, but the peak of the season is just about now, when vendors all over have a wide range of sizes. The stalks range in size from tiny, straw-sized specimens to huge stalks the width of a man's thumb. There's a persistent urban myth that the first stalks of the season are thin, then get fatter as the heat rises; this is not true at all. Stalks that grow closer to the root of the fern tend to be fatter; stalks on the outside tend to be thinner.
As asparagus matures, it gets tough and woody from the outside in. The standard advice is to bend a stalk of asparagus, which will break at the point where the tender stalk starts to turn woody. This is true, but wasteful; five seconds per stalk with a vegetable peeler will trim the woody parts and result in more asparagus. Fatter asparagus may need slightly more peeling than thinner asparagus.