How can you find which monitor has the menu bar in AppleScript?
UPDATE

Removed my original answer.  Here is a completely different approach which I prefer over my original solution.  This version actually uses “Image Events” scripting addition.  

This script will get the names of the displays connected to your computer.  The returned values of “display "Display 1" will always be “which monitor has the menu bar in AppleScript?”

property activeDesktop : missing value
property activeDesktopResolution : missing value

set displayNames to {}

tell application "Image Events"
    set theDisplays to displays
    repeat with i from 1 to number of items in theDisplays
        set this_item to item i of theDisplays
        set theName to name of display profile of item i of theDisplays
        set end of displayNames to theName
        set activeDesktop to item 1 of displayNames
    end repeat
end tell

tell application "Finder"
    set activeDesktopResolution to bounds of window of desktop
    set activeDesktopResolution to items 3 thru 4 of activeDesktopResolution
end tell

set theResultz to display dialog activeDesktop &amp; "  " &amp; item 1 of activeDesktopResolution &amp; " x " &amp; item 2 of activeDesktopResolution ¬
    buttons ¬
    "OK" with title ¬
    "Your Current Display and Its Resolution" with icon 1 ¬
    giving up after 10