In reply to girlymonkey and pete: This is always an interesting debate but there are two great difficulties to giving general advice about carrying axes or crampons.
First, we all have slightly different concepts of an average 'day on the hill in winter' primarily in terms of our perceptions and experience but also in terms of the vast range of conditions that can be encountered.
- Is it walking up hills that just happen to have snow on them by fairly straightforward, objectively safe and well established routes ?
- Is it undertaking Mountain journeys making the most of the Winter conditions to give an interesting and varied experience?
- Is it all about battling the elements and having an adventure?
- Is it just a bit of a walk in the fresh air to work up an appetite?
Second is the fact that safety has everything to do with knowledge and attitude and precious little to do with equipment. Off the back of 20 years experience I'd have a quality day out, with or without any combination of axe or crampons. Unfortunately gaining the ability to plan, select routes and make decisions on the ground is not learnt overnight.
For all that prevarication I will get off the fence and say 'crampons first'.
The truly important issue is footwear.
If you don't have good footwear (i.e. B2/3 boots) then crampons are just imperative - far, far more important than an axe. With familiarity and a defensive mentality even basic strap-on crampons along will cope almost all situations. Combine crampons with a walking pole and I would not be too bothered whether I was still carrying an axe.
If you do have a pair of B3 boots then carrying just an axe is sufficient to cope with the full gambit of conditions (assuming of course your step cutting is up to scratch). However, even with good boots, carrying crampons is at least as effective as carrying an axe.