Notice: I consider this a question about classical mountaineering. The question becomes very debatable if you include steep ice.
Do not use hand leashes on mountaineering ice axes: you attach a sharp tool to yourself which has a high chance of serious injury in case of a fall.
While there is a whole bunch of pros and cons, the one deciding factor that all mountain guides and instructors I met agreed on is the following: If you fall and let go of your ice axe, you will not get control of it again even with a leash. So the only thing it accomplishes, is that the ice axe remains close to you and will probably injure you. So no benefit but big security risk.
There are of course some situation where a hand leash will benefit you, coming into my mind are: Accidentally losing grip and falling into a crevasse (lower risk of injury due to short fall). An obvious handling disadvantage is: Loosing time when changing direction and thus the hand holding the axe. All these factors however are much less important than the security implication explained above.
Self arrest / falling:
Most importantly: Train self arrest with your ice axe (without crampons or leash). This will show you that it is very important to stop a fall quickly. All people that I have personally see slipping on a snow slope stopped quickly by self arrest after falling. People that fell for a longer time all stopped their slide only when the terrain got less steep. Two friends who slided/fell for a long distance both reported that there slide quickly became and uncontrolled tumble after building some speed. The essence: You should never become fast but break quickly - recatching a lost ice axe takes too long and will quickly be impossible.
A personal note on steep ice:
I do not use leashes either. However I am no expert and this is purely on recommendation of an expert friend. His opinion is, that with hand leashes you will be more likely to drop one as you have to fiddle with the leash when placing an ice screw. And for really technical climbs leashes attached to the harness often get into the way, which is not much of a problem in simpler terrain/classical mountaineering - there he does use them. That is however just a personal preference.
Another note on shops:
Most ice axes are supplied with a hand leash. However the mountaineering shop I go to advises all its customers to remove them before use. I asked them why they do this. Their answer was the safety implications described above. The second question: Why then are leashes still supplied. The answer is marketing. Of old hand leashes are part of an ice axe and customers might consider another ice axe if this "extra" is missing.