I never clip my axe to my harness. I have a phobia about sharp objects attached to my groin. Plus, it just seems dumb. I've got enough shit on my harness. Plus, I'm not always wearing a harness when I'm using my axe.
However, I do use a wrist leash, made from thin (9/16'') webbing. True, a loose axe is deadly, but if I loose control of my axe I figure I'm fucked anyway. I keep a pretied loop near the axe head which is convenient for giving a quick carabiner/axe belay ("stomper" belay), which I prefer to a boot/axe belay -- just plant the axe, snap a biner into the loop and you're ready.
I keep the leash long enough so that I don't need to switch wrists at switchbacks, but short enough that I don't trip on it.