Usually when a capacitor is damaged, one of two things happens.
It either becomes a short or a complete open. More on this later!
A short can easily be detected using a cheap multi-meter. You can even test for that with the capacitor still soldered on the printed circuit board (PCB)!
An open is difficult to test since a functional capacitor acts open for DC current anyway. However, a capacitor that has a relatively high capacitance, which most electrolytic caps do, behaves differently than a simple open.
If you measure the resistance of a capacitor with high capacitance, the meter shows a low resistance for a moment and then the reading goes up to infinity. This happens because the capacitor gets charged by the Ohm meter! While the capacitor is being charged, the Ohm meter thinks that it is measuring a resistor!
If you then reverse the meter probes on the cap, the same thing happens again. This time, the capacitor is charged with the opposite polarity. This should be safe since the current an Ohm meter provides is quite limited.
You may need to try this using different scales on the Ohm meter (this would effectively change the current that charges the cap) to see this effect. You need a low enough charging current so that charging takes long enough for the meter to react to.
A capacitor that behaves this way is not open. An open cap simply reads infinity all the time. So, this is a way to tell if the cap still acts as a cap or it is just open.
Keep in mind that if you are testing a cap while it is still soldered on a PCB, it may be in parallel with a resistor or a transistor. Then, some Ohm meter scales will be useless. For example, if you have a 10-Ohm resistor in parallel with the cap, the meter is always going to show 10 Ohms. All scales like 1MOhm, 100kOhms, 10kOhms even 1kOhm will be useless. And if the behavior I described only occurs with those scales for the capacitor in question, you are out of luck and you will only be able to test the cap if you remove it from the PCB.
A capacitor with small capacitance like 100pF will not behave this way because it gets charged so rapidly that the meter cannot react and only shows open. So, for a small cap, you definitely need a capacitor meter. But, Electrolytic caps, which are the ones that bulge, are used when high capacity is required.
It is extremely rare for a capacitor to get damaged and the damage manifests as a change to the capacity. I mean, as I said, the capacitor either gets shorted or opened. But, if this happens (capacity change but not short or open), the tests I mentioned, using a multi-meter, may give you wrong impressions. This means that the test I described may be useful as long as you know its limitations, and reasoning behind it.
Edit:
Believe me, I am trying to keep it short!!!!
I forgot to say that in general, there is always leakage you have to keep in mind specially for Electrolytic caps. So, if you increase the Ohm meter scale to something like 20MOhms, you may see that the reading never goes all the way up to infinity. This does not necessarily mean that the cap is damaged. It only means that the meter is measuring leakage.
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