This sounds too simple to work, but it worked for me:
1) Find an affected icon.
2) Click and HOLD the icon with the Left mouse button.
3) Press Esc while holding the Left mouse button.
That's it! This was driving me crazy. It had nothing to do with registry entries or anything else. Simply a failed Drag & Drop operation that hadn't exited. Even restarting had only worked temporarily, so I suspect my Esc key may have been stuck down or at least the system thought it was!
My particular variant of this problem seems to stem from either a Virtual Machine taking mouse focus, or a browser window (I was running a VM inside a browser window) then being interrupted during mouse action.
Thanks to @_Nige for the quick fix.