It is easier to group basic problems with common causes on the VIC-20.

Rarely will there ever be trace damage. It is almost also IC failure, and the failure of specific ICs gives common output problems.

This indicates most commonly a bad power supply, fuse, regulator, or a dead short. All in all preventing the system from receiving power.

It can also be a bad VIC chip (under the metal can near AV port).

Less common could be your TV, video cable or video circuit on the computer.

If you get a valid video signal but all black this is usually:

If you get a screen with no border and all a solid color, it's almost always a bad VIC (under the metal can near AV port).

If the user port or cassette ports are not working, this is usually the UAB3 (MOS 6552).

If the user port or cassette ports are not working, this is usually the UAB3 (MOS 6552) or the UB4 (7406).

If it isn't the actual keyboard, it is usually UAB1 (MOS 6552) by the keyboard connector.

If not the actual joystick or physical port, this is usually the UAB3 (MOS 6552).