I found a nice model for a ship from the game "Eve". It doesn't have a flat bottom, so it needs support material. But Slic3r generates several dozen tiny support pillars, and one by one they break loose from the build plate. As they get tall, the leverage of course increases, and since they're so tiny they don't have much area in contact with the bed, so they separate. Eventually the whole model broke free, turned a bit, and since I had gone to dinner it kept printing... see below. I think the main problem is the poor support material geometry, but the controls in Slic3r for support material don't seem to give many options.
How can I get better support material layout? Oh, this is PLA, by the way.
(appending to question to be able to put in the picture)
I tried a bunch of things, and the MeshMixer support, plus fiddling with several settings, got it to come out pretty well. Thanks to all, esp. @Tormod!