A "yay" thing: I finally got around to starting on that this evening!
A "boo" thing: This is a sink that is basically glued to the underside of the countertop with caulk, rather than going in from the top. To assemble it in the first place, the factory assemblers (I'm pretty sure this was a pre-made unit) cut a hole in the plywood cabinet top just a bit bigger than the sink, and then screwed some plywood support blocks onto the bottom of it to make ledges for the lip of the sink to sit on. They then sat the sink on those ledges, which left the top of the sink about level with the top of the plywood, and glued the ceramic countertop to the plywood cabinet top, trapping the sink in between.
In a reasonable world, the plywood support blocks would have been screwed on from the underside, so I could have simply unscrewed them. However, they were screwed in from the top, which meant the screw heads were trapped between the plywood and the glued-on ceramic countertop.
A small "yay" thing: The screws were slightly longer than the two layers of plywood, so I could at least see where they were by the tips sticking out of the bottom layer.
A "yay" thing: The local Lowe's Hardware Store was open until 10pm, so I could go and buy a small hole-cutting saw (imagine a tube with saw teeth on the end, that goes into an electric drill), and I was able to use that to cut around the screws and get the support blocks out that way. I could then unscrew the remaining little cut-out plugs off the screws and break off the ends of the screws.
Another "yay" thing: I had been wondering how to do this without having the sink basically drop on my head when I removed the plywood support pieces -- and how to hold the new sink in place while the caulk dried. Conveniently, the internet provided an answer -- put a 2x4 scrap across the top of the opening, and drop a pipe clamp from there down through the hole in the sink to hold the sink up. This worked perfectly, at least for getting the old sink out, which is as far as I got.
A "boo" thing: It then took me a half-hour of looking to find my caulking gun, because my tools are currently in semi-packed-up disarray due to another project (namely, cleaning the space where they go and creating organizational stuff like pegboards so eventually I can find them all easily).
Another "boo" thing: Once I found the caulking gun, I discovered that my unopened-but-old container of silicone caulk had hardened in the tube.
So, tomorrow I go get more caulk, I guess, and see if I can get the new sink in place.
Crossposted from Dreamwidth (original here), with comments. Comment here or there; comments here will eventually be duplicated to there.