Easiest way to find out if user has either Windows 7 or Vista (through telephone support)?

Press CTRL + ALT + DEL it says so in the bottom and most people are aware of this shortcut (especially in corporate environments where people have to press CTRL + ALT + DEL to login).

vs

Press Win+Pause (if you can explain to people where those keys are).

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For reference, here are screenshots of the resulting dialog under various versions of Windows:

Windows XP

Windows Vista

Windows 7

The quickest visual way to tell the difference between Windows Vista and 7 is the taskbar by the clock. In windows 7 there is a box that can be clicked to show the desktop. Vista:

Seven:

This might look different if a user has changed the theme or something weird, but if they can change a theme, hopefully they should be able to manage one of the other methods suggested.