Wow... Interesting popping. To me it sounds like a PSU problem or you got a very noisy AC Voltage. Check the PSU...
To check the PSU, put the black wire from the multimeter onto the metal casing of the PSU OR one of the 2 black wires in the harness and make sure it makes contact with the pin in the harness. Of course - turn on the LC III...
For 5V - put the red wire from the multimeter to the red wire in the harness. The Multimeter should read anywhere from 4.75 to 5.25 Volts, preferably at 4.9 to 5.1 volts. Looking at a LC PSU right now, there is no Red Wire, but it should be the Orange Wire that is 5V.
Now I forget which is which, I think the yellow wire is 12V and the Blue wire is -5V but it might be the other way around. Either way, when you measure those wires for voltage, you should get 11.75V to 12.25V for 12V - best at 11.9v to 12.1V; and -4.75 to -5.25 Volts, preferably at -4.9 to -5.1 volts.
The problem is this, if it is a cheap digital multimeter, you may get a rock solid steady reading of these voltages as the sampling rate might be low. If it a good multimeter with a higher sampling rate, then if you should get a reading that flutters about every seconds or so if there is a problem with the PSU. This is why I double check with an analog meter, it there is a flutter or spike (which I suspect here), you can see the needle shake much more visibly. A good PSU should have rock solid readings.
For me, this is either a PSU problem or a Noise in the AC problem. Working in schools, I seen a bad florence transformer muck up old Macs sound by adding spikes in the AC Line. When the transformer was eventually replaced, the offending noise in the Macs went away.