Identifying the Problem
I’m assuming you have some technical knowledge and are competent working on computer electronics, otherwise take your computer to a repair shop. Take note of the Terms and Conditions of this website and the disclaimer on the following page.
The most obvious sign of PSU failiure is that the computer will not switch on. Try a different power lead (the cable connecting the computer to a power outlet), and a different wall socket (directly into the wall, not via an extension or multi socket as these may be defective). If you can’t hear the fan going round at the back of the computer, it’s almost certainly the PSU. If there is a switch on your PSU, make sure it is on. If that isn't the problem, switch the PSU off, disconnect the power cable, open the case up and unplug and replug the motherboard power connectors. Sometimes they can get dirty or oxidised, and this should renew the connection and gives you a chance for a visual inspection.
Erratic behaviour at startup - multiple beeps, won’t boot every time, frequent crashing - can all be symptoms of a failing PSU, but may also point to other problems such as a defective motherboard or graphics card. Motherboard manufacturers usually supply a lookup chart for the beep codes that might point you in the right direction.
Grinding or squeaking are usually indicative of a worn fan. If it is the fan in the PSU, replace the PSU. I’ve replaced the fans before, but it can be awkward and if the fan hasn’t been cooling, the PSU has been overheating. It will fail soon. Also, the inside of a power supply is a very dangerous place, able to contain high voltages despite having been disconnected for signifcant periods.
If you have and can use a multimeter, you can test the PSU. Disconnect the power supply from the motherboard and drives, etc. To switch it on, you’ll need to briefly short out the power switch terminals. They’re usually the green and any black (ground). The excellent Russian website, pinouts.ru has a guide for the expected voltages at each pin (http://pinouts.ru/Power/atxpower_pinout.shtml). If any pins fail the test, ditch the PSU and buy a new one.
An inexpensive power supply tester is much easier to use for those who aren't electrically inclined.