There's definitely pros and cons to using different kinds of leashes or not using one at all. I can say that I chose to use a stretchy leash tethered to my waist/hipbelt because I wanted to be able to easily switch hands with the ice axe when we were doing switchbacks (so that the axe is in the uphill hand). My ice axe came with a wrist leash but it's not long enough where I could keep it on one wrist but use the axe in the other hand, so I chose to forego it.
My view is that regardless of whether you're leashed or not, it's bad if you are moving and your ice axe slips out of your grip. I prefer the peace of mind knowing that during those moments when I'm not needing the ice axe (e.g. when I've braced myself in the snow during a break), my ice axe isn't going anywhere. If I'm falling down the mountain and the leashed+flailing axe is stabbing me, I've already done something really wrong and that needs to be avoided. Equally bad is if I'm falling down the mountain without the ice axe. But let's not get into this great debate
