My SMB shares won't mount under Catalina! What's wrong?
It turns out that the version of Samba (3.0) running on the server only does SMB1, whereas Catalina requires at least SMB2 (which didn't come in until Samba 3.5).

Caught between a rock and a hard place.

Given that the Samba box (1) is now only being used, at least on any sort of regular basis, by me, making it gross overkill, (2) is literally doing nothing but serving SMB1 and consuming power, and (3) is a 15-year-old Dell 400SC, I'm probably going to replace it outright. Possibly with a thumb drive.