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Thanks, guys (how did I know that you two would be among the first to post answers?).
Tried safe mode, fo3nix and it didn't seem to make much difference. So then I tried latest-trunk, but that decided that almost all of my add-ons were incompatible. I then tried disabling add-ons one by one, also without effect!
Finally I tried uninstalling add-ons, and discovered that most of the problems went away after removing VRE Toolbar.
It's too early to say if it's solved all the problems, all of the time, but it's definitely an improvement!
Chris, are you using VRE toolbar?