Upgraded to Windows 10 a couple of weeks and just noticed that almost all of the icons on my desktop have a small red X in the lower-left corner. The recycle bin and some documents seem to be the exceptions.
Don't know if this occurred right after the upgrade or more recently.
Any ideas on what this is supposed to indicate?
Sounds like the Icon Cache needs to be rebuilt.
According to Windows 10 Forums,
The icon cache can sometimes become corrupted causing the icons of files to display incorrectly or distorted. When this happens, the icon cache needs to be deleted to reset and automatically recreate it.
There is also a tutorial there to walk you through it.
Right click desktop and choose "refresh". It will redraw all your icons and remove the gray [or red in your case].
Your Windows Shortcut icon likely got changed.
Try resetting the icon back to its default. In the following key, set the string value 29 to %windir%\System32\shell32.dll,-16769: HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Explorer\Shell Icons.
Logoff for the change to take effect.
Source: Option 3 of Seven Forums - Shortcut Arrow Change, Remove, Restore
Note: While written for Windows 7 and 8, this should work in Windows 10 as well.