*tears out hair*
Fortunately the leak is incredibly tiny, basically a pinprick in the copper tubing. But my attempts at staunching it (superglue! duct tape!) were for naught, so now it's got a little bowl under it which it probably wouldn't fill even if I let it go all night, but it's the PRINCIPLE of the thing. Well, that and I know it's going to be the first thing the plumber asks me when he does show up. And if I start letting household fixtures go walkabout there's no telling what I'm going to wake up and find missing.
EDIT: On the bright side, I figured out why the ice maker wasn't working right. Har har har.
EDIT 2: Mystery solved, I think. J. got home and pointed out I wasn't losing my mind (further). Yes we did have water when we moved in before we got the fridge, but no the line doesn't have a shutoff valve: it has some pressure-based ball-bearing autopluggery end valve thing that caps off the end when you disconnect the fridge. Which is all very well and nice but does BUGGER ALL when the line itself has a hole in it!
Ah well. At least I'm not completely mad and/or incompetent. And I'm being repeatedly amused by Palin sneaking back there to try and drink the water out of the bowl and being surprised EVERY TIME by the pipe squirting water in his ear. He is an adorable little spazzmo, but he is not at all good with the short-term memory vs. shiny things.