Capacitor Failure Methods
The scope of this page was poorly defined and this information should most likely integrated into Capacitor Types as the failure methods will be more dependent on the type of capacitor.
Capacitors can fail in many different ways. Ideally they gracefully fade over time as their capacitance slowly decreases and you are able to notice the change and replace it. But they don't usually have the kindness to do that. Some of them fail rather spectacularly if they have shorted internally and will be very obvious to find. Others may heat and erupt spraying all over. Some may "grow" out of their cases through the top. But probably the most likely fail is from leaks that shorten the life of capacitor and possibly causing damage to the circuit board.
Now if your capacitor hasn't failed visibly and you don't know exactly which one it is you are in a bit of a tough spot. There are many devices that are sold that claim to be able to test capacitors without removing them from the circuit board, but that is not always be true. So you can't rely on a diagnosis from that alone. You are likely going to have to take some diagnostic steps to isolate where the problem likely is and start checking individual capacitors from there. It is best to use a multi meter to check voltages against a schematic for known values, but if that isn't available then you are going to have to make some assumptions. Capacitors have a rated voltage that the power they work with should be less than, you can try to find capacitors that aren't getting a voltage near their rated value and then looking at where the voltage would be coming from. But this isn't fool proof as not all capacitors are used at their full voltage all the time(like an audio amplifier as an example).