The Spesh Tricross bikes often pop up on this forum as a "one bike to do all" - it can do off-road, on-road, take mudguards and a rack, handle the winter, commute, do some light touring and is, apparently, reasonably comfortable. You can also race cyclo-cross
Of course, it might mean that it isn't particularly great at any of them (save the cyclocross, for which it is designed

). I have a friend who rode Edinburgh to John o' Groats on one last year and he says it was fine.
Singlespeed is, of course, another matter, about which I know little - other than I would love to try it
Bit stuck really on what bike I should be looking at.
Budget, budget, budget

, read the
My Advice to Newbies thread and see if that helps.
let us know how you get on
