Awesome thread, definitely worth discussing.
Statistics: the NPS has the best stats. They don't turn up anything meaningful. Mainly because not enough solo hikers are out there dying to provide us with good data, but also, because the experience they've had varies wildly from newbies to the most seasoned vets who simply had a bad day in the mountains.
Climbing statistics are more telling, for climbing routes at least, and they can shed some light on the subject. Solo climbers tend to be more competent, more confident, and more able to assess risk and tuck their ego away (while they're climbing at least, all bets are off back at the bar). As such, at least from glancing through the past 10 years of Accidents in North American Mountaineering, few are true "solo" accidents.
Personal experience climbing: I've gotten myself in stickier situations with groups than I ever have solo, with one or two exceptions. (Especially when Alan Dixon is my climbing partner. We somehow have this thing for climbing and descending in the dark.)
Now, translate all this to backpacking: I think that generally, solo hikers in remote mountainous areas are generally more aware of the risk than those folks that go in groups, and barring the occasional disaster by which boulders pin limbs requiring you to chew your way out of the wolftrap, solo hikers are able to suppress their ego some because no one is watching and they tend to play it fairly safe. However, for more experienced backpackers, going solo means more risk because they may do things that they wouldn't do with others less experienced than them.
I don't think twice about going solo, although I used to: Freedom of the Hills and BSA theology ingrained the awful risks of going solo and it took me a while to deal with that mentally. I do think twice, three, four times about going solo in grizzly country, and in avalanche terrain (winter) and am far more aware of the risks. As such, I try to be extremely careful in both of those situations. I can avoid avalanche conditions in the winter. I can't avoid grizzlies, because you never know when one is around the corner.
The benefits of solo for anyone far outweigh any risks: finding emotional solitude now and then is good for your health. Going solo doesn't mean taking big risks. A simple 3 mile overnight hike close to home on an easy trail is a whole world better than staying at home worrying about what happens if you break your ankle while hiking alone.