I am trying AVG Free as well. It has already scanned the registry, processes and system directory and found nothing.
I am doubting a virus, since even pressing Alt+Tab to change between programs temporarily fixes the issue. I am thinking it is some sort of mapping problem. I think that the OS sometimes thinks that the taskbar is offset somewhere.

This advice looked good.
http://forums.techguy.org/windows-nt-2000-xp/566349-cant-click-desktop-taskbar.html
It recommends creating a new Admin user account and using that instead.

The new user account did not SEEM to have the problem (it is intermittent so it is difficult to say)

But before I could copy my old user profile
Microsoft Corporation
wholesale to see if the bug comes with the copy, the taskbar became unresponsive again.

Somehow the new account seems to be faster. Maybe I will stick with it.