Generally I use a pair of trekking poles. I prefer trekking poles to a single stick/staff for a few reasons. First, I like the hand straps that most trekking poles use, which mean you don't actually have to grip them at all, you can just let them swing on the straps. Second, I like being able to maintain three points of contact when crossing a river or on ice. Third, I appreciate the redundancy of having at least one pole if one were to break. Finally, I usually use my poles when I tarp camp as part of my setup.
During the winter, which lasts quite a while around here, I add larger snow-specific buckets to the trekking poles. Walking through deep snow and across ice seems safer with two poles.
These are my thoughts on this matter.