But given that other USB devices work here — what sorts of other USB devices?
I have 2 external hard drives that work on the very same USB port and an external blu-ray drive as well.
Are there any add-on anti-virus or anti-malware "tools" installed?
None. Never have.
To check the USB bus, remove all hubs and all other external USB devices from this Mac, reboot, connect one of the thumb drives directly to the Mac, and try again.
Everything re-connected except for the thumb drives. External hard drive and Blu-ray drives popped right up but no USB flash drives.
Try a different and newly-created login, and see if the flash drives work there.
I used the Guest login and the same thing, everything works except the Thumb drives.
To check the local OS X installation, see if booting this Mac from the recovery partition (if one is present)
None is present.
or booted from another (external) OS X disk allows access to the thumb drives.
I have en extern HD that is blank, but I do not know how to install Mavericks on it then run it.
Is there anything in common among these failing USB disks? Same brand and model? Or a mixture of different thumb drives?
They are all FAT32, other than that no I have Cruzer, Kingston and PNY brands. 2GB 4GB and 16GB.
Are the OS X systems involved both similar configurations, or are they very different versions?
Mac Mini is Mountain Lion and has no issues with the thumb drive.
Intel iMac is Mavericks and my Mac Book Pro is Mavericks. Neither one of them can mount my USB sticks.
Thanks for your complete reply, you had some items that I did not try but unfortunately it did not help yet.