THE MOST COMMON FAULT WITH A ZX81 IS A FAULTY ULA CHIP
 
COMMON SYMTOMS OF A FAULTY ULA
 
1. A very dark almost black screen with a cursor at the bottom left which 
   is very dark, blurred and just about visible.
2. A completely black screen with no visible cursor at all. (This can also 
   be a power or other problem).
3. A White screen with sort of black zig zag lines that appear to move up 
   or down screen.
 
The above are only the most likely symtoms of a faulty ULA. I can't guarantee 
this will fix your ZX81
 
 
The ULA controls the Display and Tape I/O (input/output), this chip normally 
runs hotter than it was designed to do, and so fails much sooner than most of 
the other components. This ULA was also only ever produced for the ZX81, and 
went out of production about 18 years ago, and so is not only the most likely 
component to fail, but until now was the hardest component to source.