It appears as though a significant number of 1.25 Ghz eMac users are suffering from logic board breakdowns that cause video to become distorted. Although Apple's included diagnostic utilities indicate that all hardware is operating normally, opening up the machines reveals bulging and cracked capacitors.
Most of the issues are occurring out of the eMac's 1-year standard warranty, and Apple has yet to set up any sort of extended repair service (although some are having limited success getting their machines repaired for free by noting the Apple discussion thread).
Raw Data:
Apple Discussion Board
Macintouch User Reports
(more reports in forum thread)