In reply to Daniel Sutherland:
I've done a lot of running on snow. Inov8 mudclaws work well on most snow - I always have a pair a half UK size bigger than my normal pair, for winter running, so I can wear a warm pair of socks and on higher mountains some sealskin socks (don't keep your feet 100% dry but MUCH warmer - until they get trashed!)
When it gets too steep or icy, kahtoola microspikes are the business - lightweight, quick to put on and off, and they don't snag on grass etc as much as crampons so you can leave them on longer. They also allow you to run very naturally (unlike crampons). I'd disagree with IanRUK about not using them on mountains - I've used them on many Munros.
Rarely do you need any more than that but if it gets very steep, icy and/or very windy then light flexible crampons are required. I have some old 8 point walking crampons that I use which fit fell shoes well, were very cheap, but don't go compact for storage. Thinking of getting some kahtoola running crampons. Suspect the steel version is best for UK conditions (if you need crampons instead of spikes, it's probably rocky as well as icy). You can run in crampons, but not as well as microspikes.
I use a BD raven ultra for an axe, with a simple dyneema sling for a leash. Not quite as light as the CAMP aluminium axes, but a proper axe, will stop you in tricky conditions (I had to do ice axe arrest on Friday in high winds on ice on snowdon!) and also great for walking / general mountaineering.
Post edited at 09:42