Hello
Earlier on, when I closed down my PC, it would occasionally do all the right things, screen went black and all seemed OK, but after a while the Mint logo appeared on the screen and the computer was actually running. And it kept on running for hours with the logo on the screen until I stopped the computer forcefully by keeping my finger on the power button. Now this happens practically every time. This cannot be very healthy for the computer, could this be a sign of something more serious awaiting around the corner?

you may try to execute shutdown or poweroff from a terminal (as root).

How are you shutting it down? Using a menu on the Mint desktop?

I've had to do that on occasion (and it's not all that predictable when it's going to happen). I've noticed that, during those occasions, the final stages of shutdown take an inordinate amount of time with a lot of disk activity taking place. It never takes several hours, though, only minutes. Are you seeing a large amount of disk activity during shutdown? (This appears to happen sometimes on my old laptop w/o and drive activity LED. Usually when I'm in a hurry. [heh] )

Well, it's not a good practice for the health of your filesystems. I'm guessing that you see boot messages (or they appear in a log file under /var/log) notifying you of fscks that are being performed due to the unclean shutdowns. Look for messages like "Starting File System Check on /dev/disk/by-uuid/blah-blah".

I'd try:

• getting rid of the startup/shutdown eye candy that distribution maintainers like to add to the boot/shutdown processes. The messages that stuff hides can be very informative.
• don't shutdown via the Mint desktop menus---just logout
• Switch to a console (Ctrl-Alt-F1, for example), log in as root, and shutdown from there. Pay attention to the messages that scroll by. I've never tried it before but you might try:
  
  Code:

  # shutdown -h now 2>&1 | tee ~/shutdown.log

  I'm not sure what that'll collect because, at some point, shutdown will kill the tee process. Something's better than nothing, though.

HTH...

Wonder if some /var log might show what was stopping shutdown.

It is possible some update has caused a change (for worse)

Guess I'd try a stock live cd/dvd/usb to see if it also fails.

Solved (sort of)
A fresh Mint 19.3 installation solved the problem.