If you know about a pattern that you don't see here, let me know and I'll see about adding a section covering it.

4-in-1 rounds

Square patches of chainmail are nice, but sometimes you need to go around corners or make something cylindrical. For these projects, you need rounds.
European 6-in-1

So-called because each ring has six others passing through it, European 6-in-1 is sometimes referred to as "king's mail", though I personally have seen this name used to refer to at least three other unrelated patterns. It's a much more dense pattern than 4-in-1, and when made with 14-gauge wire on a 3/8" mandrel, only bends easily in one direction. Due to the tight spacing, it's considerably more difficult to make than European 4-in-1.
Double-link 4-in-1 (8-in-2)

I first made this pattern while just playing around one day. Like European 6-in-1, it's only really flexible in one direction, and is a very dense pattern. I still use my original patch of this pattern as a keychain, as the weight of it is enough to keep my keys from falling out of my pocket while riding my bike. Due to the density and close spacing, I use a different method for this pattern than for 4- or 6-in-1.