In reply to Lisa_K:
I'm going to respectfully disagree with Mr Forster and say that an ice axe should be more commonly used than crampons in winter in Scotland.
Crampons are needed (for walking) when the snow-pack is sufficiently hard that boots do not sink in enough to give decent grip. That is relatively rare at low altitude, but quite common on the tops where the wind has helped compact it.
An ice axe is needed when the ground is steep enough and the snow hard enough that it would be difficult to stop sliding down it without one.
It is quite possible to find snow-slopes hard enough to need an ice axe, but not hard enough to need crampons. It is also possible to find snowy areas hard enough to need crampons, but not steep enough to need an ice axe. In my experience, the latter are much rarer than the former in Scotland.
To answer the original question, a walking axe should have a straight or very slightly curved shaft, a decent sized adze and a good spike at the bottom which easily pushes into semi-hard snow. A rubber grip on the lower part of the shaft is helpful on steeper ground, but unnecessary for pure walking. A leash is probably a good idea, though won't get used all the time. The Grivel Munro comes to mind as a good example.
Ice Axe length is a controversial subject, I know people who swear you will die if you use more than 55cm, and some who refuse to use anything shorter than 65cm. A shorter axe is easier to self-arrest with, a longer axe provides more support and prevents falls. My 'rule of thumb' is that the spike of the axe should be level with the ankle-bone when held as for walking.
Hope this is helpful.
Jack