Where has my external thumb drive gone?

For external drives of any kind to show up on the Desktop, you Finder preferences must be set to allow it.

Click on the desktop until "FInder" appears next to the Apple menu, then click on "Finder":

Selecting "Preferences" will show checkboxes. Make sure "External Disks" are checked.

Thumb drives can and do fail as often or more often than external hard drives.

Try unplugging the thumb drive, restart your Mac and then plug the thumb drive back in.

If that doesn't help, try testing the thumb drive on another Mac or PC.