how to figure out noisy fan?

So my computer has like 10 fucking fans inside of it.  Maybe not 10 but a lot.

One of the fans is making this noise, it happens randomly and when I don't suspect it.  I suspect its either dust, or maybe some of my shedded long hairs got in there, or maybe the bearings are going loose.  It's a very faint sound, I only hear it because my computer is right next to me at ear-level (another question -- if it's that quiet, is it still something to be concerned about?).  It sounds more like a faint "ffshwffshwffshwffshw" instead of a more abrasive rattling or grinding.

Since it doesnt make this sound all the time, I'm not sure how to pinpoint which fan it is.  Last time I opened up my computer and waited an hour for the sound to show up, the sound didn't come around.  It could be the CPU fan, the video card fan, one of the main circulation fans, the power supply fan, or some other fan in there that seems to be on its own (its on the plastic "door" side of my computer, not where the motherboard/etc are... no idea what that's for).

I'm thinking of buying a stethoscope so I can quickly use it the next time the sound comes up... I'm not sure if that'll be useful, though....

OP here

Okay just now my computer emitted a couple of very faint, short-lived "bzzzzt" sounds.  Not even prefaced by the "ffshwffshw" sounds I mentioned before.  And now there is a noxious odor in the room.  I don't see any smoke and, well, my computer is still working since I'm typing this.

What the fuck is happening?