For the past week or so, the standard "Shut down" dialog appears at random, with no apparent cause.
I upgraded from 12.04 to 12.10 and it still continues. It's unbelievably irritating.

It can appear when I'm just typing into a text document, hours after the computer booted, or it can happen three seconds after I log in and haven't had a chance to interact with the desktop.
If I press "Cancel" the dialog may reappear within a few seconds, or a few hours. Sometimes the laptop just powers off instantly a few seconds after pressing "Cancel".
When this dialog appears, I don't see anything weird in /var/log/*log and I don't see any suspicious processes running.
I have managed to capture this by monitoring the dbus session bus, when the dialog appears:
method call sender=:1.6 -> dest=org.gnome.SessionManager serial=285 path=/org/gnome/SessionManager; interface=org.gnome.SessionManager; member=Shutdown
method return sender=:1.0 -> dest=:1.6 reply_serial=285
But that gives me no clue as to who is requesting a shut down.
The physical power button (on this Thinkpad T420s) is not mapped to the shutdown prompt, and pressing it does nothing.
I've also seen this random shutdown behaviour if I boot the laptop, then kill lightdm so that no X server is running. At some point, the computer may sporadically say it's going down for shutdown. Then it does.
Any ideas what this could be, or how I can debug further?
Update:
This also happens with a Ubuntu 10.04 live CD and a 12.04 live USB drive.
However, I swapped out my SSD and put in the original HDD and I was able to install and run Windows 7 for a while without problems.
So, since this problem appeared recently and occurs on multiple OS versions, I'd guess it's potentially a hardware problem. But given that I can't see what (haven't seen any phantom key presses in showkey) is wrong, and because Windows works, I don't really have a good case for calling in my Thinkpad warranty...