Our stairs are completely enclosed - walls on both sides. I've just removed an ugly and flimsy handrail and the stairs look so much better and more open without it. If we replace it it would be something much more flush to the wall (the old one stuck out on brackets) so perhaps the space issue wouldn't be so bad, but it seems like a huge faff to replace if we don't have to.
I've found regulations that say that you need one handrail if your stairs are less than one metre wide and two if they're more than one metre wide, but how applicable are they to private homes and who is going to be checking?!