I'm running Windows 7 x64 with cloned displays: a Samsung P2770HD connected through HDMI and another Samsung SyncMaster 2243 connected through a VGA cable.
The issue appeared on the designated secondary display, which is automatically set based on the Preferred Display settings in Catalyst CC. So if you set one of the two a preferred display, you will encounter this annoying line on the other one.
I got rid of it from the desktop by disabling Aero for Windows. This doesn't help games, which still show a visible tearing. I have forced the Vsync On and enabled Triple Buffering from CCC but to no use. Also note that this is by no means a vendor exclusive issue, as it was encountered on both ATI and Nvidia, so I presume vendor drivers can be ruled out.
If anyone has any clue about this irritating problem, feel free to contribute because I searched through almost all the threads that deal with "the line" (it's like an abnormal warping of the video signal, probably slowly braking the fabric of the universe, so this is URGENT) and people don't have a permanent solution. Based on what I've read and seen, I think this has to do with Windows handling duplicate displays.