You're going to have to do a thorough inspection and (possibly) destructive testing. But, since the PSU isn't operating properly, and shouldn't be powering your PC in this condition anyway...
1.) Turn off PC, and remove AC power cord from the PSU;
2.) Press power button on PC (to discharge PSU caps.);
3.) Unplug 24-pin cable;
4.) If you can work on it with the PSU installed in the case, do so, if not, remove the PSU;
5.) From connector side, look into each connector slot. You are looking for any of the metal connections that look out-of-place, set further back from the others, or scorched or physically damaged;
6.) Turn the connector back over and hold it in one hand. With your other hand, pull firmly and quickly on EACH, INDIVIDUAL WIRE NEAR THE CONNECTOR. with about as much force as you would pluck a hair. If the pin slides out, BE SURE YOU NOTE THE ORIENTATION OF THE PIN!. Inspect it for damage, and check the lock tab. (Little ear or pin that should be sticking out the side of the connector about 3/64'' or 1mm). if it isn't, or it's bent, or it's flush, then you will have to fix it;
7.) If the tab is flush or not sticking out far enough, then slide a very sharp, thin knife into the open end, and pry it lightly out. DO NOT BEND IT VERY MUCH (past 45 degrees, it will probably snap off, then you are REALLY screwed!!);
8.) Slide it back into the plastic 24-pin ATX connector in the same orientation that it came out in;
9.) Once reinstalled, pull firmly and quickly on that, specific wire near the connector. It should NOT come out (again);
10.) Repeat for ALL WIRES IN THE CONNECTOR;
11.) In the event that the tab ''folds'' when the connector comes out, you will have to VERY CAREFULLY straighten it with a pair of needle-nose pliers;
12.) In the event that the wire comes out of the crimp connection...DO NOT WORRY! as this was the problem, and you can't fix it anyway! (Unless you have wire cutters, wire strippers, molex crimp connectors and a crimper?);
13.) As for the motherboard connection...with a bright flashlight, inspect each pin slot for melting, burning, scorching of the plastic shroud, and/or misaligned or loose pins;
14.) If you find any of these things, I can't help you!! The motherboard is TOAST!
If, after reading this, you do not feel comfortable tackling it, then don't. But, just a word of warning, a poor connection like this will lead to faults with the Motherboard, CPU, memory...basically it could permanently damage EVERYTHING THAT PSU IS HOOKED UP TO, also it could lead to further physical damage to the 24-pin connector(s), as I described above.