I'm not a hardware expert, but I suspect a dry joint somewhere, or a flakey video ASIC. It runs normally so long as you don't touch it, but if you rest your hands on the space below the keyboard, or you tap too hard on the trackpad, then you get random regular grey patterns on the screen and it stops responding. Services still work, sometimes, so I can still SSH into it and so on --and sometimes they don't. Or, at least, sometimes the IP stack goes down too. Perhaps the Ethernet is near the video on the motherboard?
It's two years old, and came with a one-year warranty. I've hammered it continuously since I bought it, so I don't think I can complain.
I've ordered a new one. It should arrive on Friday 10th.