In an ATX PC Power Supply you have 1.2+, 3.3+, 5+, 5-, 12+, and 12- (and possibly 3.3-) voltages. A potential need of monitoring 7 different voltages! While I do not disagree that faulty PS could shorten the life of items attached, it seems like more work than it would be worth. I'd prefer to spend the money for a known accurate and reputable PS...probably the least expensive option.
If you wish to pursue this, you could buy a bunch of cheap multi-meters from a discount tool store. Of course to be practicable you might consider getting an Arduino and program it to monitor the voltages. This forum post shows how to monitor 12, 5 and 3 volts. It should not be difficult to expand the program to monitor additional voltages. With an Arduino UNO you are limited to 6 analogue inputs, so you would need the Arduino Mega 2560 if you need to monitor more.
Furthermore, you probably want to log not just monitor so you'll want to replace the LCD shield (a board which fits on top of an Arduino) with a data logger shield, such as this one.