This is an old post, preserved for reference.
The products and services mentioned within are no longer available.
One common source of failure in consumer electronics new and old is memory. Often indicated by random failures, video RAM by garbage on the screen, and sometimes just blank screens. Usefully, this is often accompanied by the chips getting hot.
Most are in the 25-30 degrees range, and are fine, but there are three from 37 to 45 degrees which are likely to be faulty. A quick bit of desoldering and some replacement chips and a run through with the Commodore 64 diagnostic cartridge I put together last year, and another Commodore 64 is restored to life.