I probably missed something ...
I upgraded from 2.6 to 3.0 with no issues; ver on admin page read 3.0.
But I just uploaded 3.1 and ran utility/update (success), and it still reads 3.0.
Maybe a browser caching issue? To be sure, check the file /environment.php for a line like define('APPLICATION_VERSION', '3.2');
Wow. Either I re-uploaded 3.0 or I told Filezilla not to overwrite, but either way, it was 3.0. It's 3.1 now and all is well.
R_J, seems you are always coming to my rescue; thanks so much!