If wearing compression type shorts/capris, commando. Advice I received from a couple of running coaches.
Husband likes Nike dri-fit boxer briefs.
I found fit to be most important. Fabric type didn't seem to matter as much - but insist weight and my underwear was a little too big, and it would chafe me. (TMI) Recently started going commando, quite nice.
I've been wearing Fruit of the Loom micro-mesh. They are a good fit and breathe a bit. Haven't had any chafing, even at long distances.