Problem: small 12V desk lamp doesn't work. Started poking around inside it and found that the transformer primary showed an open circuit. This was a bit odd, since it didn't show any signs of overheating. So, I pulled the transformer apart (and by apart, I mean apart; remove the laminations and unwind the coils) to see what the problem was.
The primary is protected by a thermal fuse which cuts out at 125°C (these transformers can run pretty hot). It's a type M30 made by Joint Force Metal Research & Co. Ltd. (http://joint-force.net/jk/zycp.htm). Unfortunately, it's a non-resettable fuse - there's a small piece of low-melting alloy inside which melts and breaks the connection. They are available easily on EBay (search for M30 thermal fuse), but why not try replacing it with a resettable cutout?
China to the rescue, because there's nothing cheaply available in the West. I eventually determined that the thing to search for is a "bimetal switch". EBay has tons of them, but I got mine from Uxcell (http://www.uxcell.com or EBay user "shinyempire"). They're available in 5°C increments - search for bimetal switch and the temperature (e.g. "bimetal switch 115"). Although the original switch was 125°C, I went for a 115°C version to make life a bit easier for the transformer. There wasn't space to wire it in to the windings, so I brought the leads out separately.
After some careful reassembly, it works as good as new! I couldn't quite get all the E laminations back in - they're assembled under force and spring out to fill the bobbin, so I ended up with one left over, which shouldn't make a lot of difference. It's hardly worth the effort and wait for a lamp that only cost about £10, but it's very satisfying fixing something that was never meant to be fixed.