LG LCD TV has a thin horizontal black line across screen when on?

Could be a bad gate driver IC on the panel or an inverter IC failing to deliver proper power to the gate driver intermittently.

Another scenario could be a flawed LVDS cable, in either case the TV will need professional servicing, if it is still under warranty take it back.

This can be caused by an error in manufacturing, bad inverter IC on the gate drive board, or if the TV has been moved aggressively or something knocked up against it.
if you want to have an idea of what is going on and what it would cost to repair, here you are a link of a technician service www.tvrepairjacksonvillefl.com

LCD TVs are extremely delicate around the panel area.....

albert
November 2012