Far as I can tell, the issue is ice dams. It looks like the front gutters have frozen solid, and they don't get much sun; the upper roof, however, does get a bit. So the melt comes down the roof, hits the gutter, can't get through it, and backs up into the flashing. Along most of the front, this isn't too terrible: it comes out the back of the flashing and runs down as icicles. Unfortunately, though, one of the things under that flashing is that lovely front bay window of ours.
So the past few days have been bucket-focused. Fortunately, the drips are well-behaved: there are about five distinct spots where it is coming through, and we've put big buckets under all of them. No significant damage from the drips yet, but the situation can't persist in the long run.
If the weather predictions come true, I suspect the problem will ease on Sunday -- it should get warm enough to melt the gutters, at which point I expect the worst of the problem to go away. But come spring, we'll have to look into a better solution. I suspect that will involve restructuring of that gutter, but I'm not sure of the details yet.
Oh, well. Compared to the old house (whose roof got ripped off in a hurricane, producing one of the worst weeks for me ever), this is pretty minor stuff. It's still a nice house, just showing the warts now that we've been in it a while...