Computer noise - squeaking sound from computer

You will most likely have to open the case to try and find where its coming from. 

Using something like a stiff piece of cardboard you can momentarily stop the fans on both the video card and power supply.  That should identify or rule out them out as the squeaker.

c) mouse

Fans in power supplies can make a squeaking type of noise, which is not good, is usually accompanied by a vibrating type noise as the fan becomes unbalanced, as it indicates that the fans bearings are worn out or caked with dirt, which results in the fan not working adequately or at all, resulting in an overheating problem which results in your computer not working.
Its easy to tell, its usually the sort of noise you hear from the other side of the room, and is more constant like a drone, that cuts in and out when it feels like it, before eventually becoming constant, before the the power supply literally fries itself.

850W is rather overkill unless you plan to use the PS to fry things on too. I cannot see sigs from a phone, but I highly doubt that is required (or desired) for your machine.

In addition to the components in my sig, the computer also has an nvidia gtx970, 2 ssds, and 3 sata drives.  So I like to aim high when it comes to power.  Sounds like 850 is not considered high anymore!  A 1200 would be mega expensive here in NZ though!

I've been coasting along for a month and a half now, not worrying too much about the squeaking until today, when I turned on my speakers for the first time in ages.  Then I heard the squeaking amplified over the speakers.  Yuck.
 
The speakers are connected to my M-Audio Delta-44 PCI card.  So are my headphones; if I turn them up REALLY loud, I can hear the squeaking on the headphones as well (but I never turn them up that loud normally, so the squeaking is not an issue when using the headphones).  Anyway, this means the Delta card is somehow picking up the squeaking sound and amplifying it.
 
My other sound card (Xonar) is not picking up the noise.  This is strange, since the Xonar is cheaper, and is positioned even closer to the power supply than the Delta card.  Hmm...
 
I suppose I could try moving the Delta card to a different PCI slot, but I am worried about creating other IRQ conflict (or are those a thing of the past?)  Also, my case is pretty crowded, so I don't know if there is a free PCI slot in there.  I'm not very confident that moving it to a different slot will help much, since it's picking up the noise when it's 2 slots away from the PSU (and the Xonar is not picking up the noise, despite being adjacent to the PSU).  This, plus the fact that I don't enjoy moving hardware around, makes me reluctant to experiment by moving the Delta card to a different PCI slot.
 
Are M-Audio Delta cards known to amplify computer noises more than the average sound card (e.g., the Xonar)?  Or is there a chance that certain PCI slots are more susceptible to picking up noise than others, regardless of proximity to the PSU?  What else could explain why the Delta is picking up the noise by the Xonar is not?
 
Obviously, if I can eliminate the noise, the Delta-44 would have nothing to amplify (for now)... but if the noise turns out to be something like the motherboard or the graphics card (expensive!) rather than the PSU, then I am unlikely to ever fix it.
 
I'm a bit frustrated at this point, as this noise is discouraging me from using the speakers, which I would like to use for mixing my songs (instead of headphones). 

I've been doing some more tests, and I'm pretty sure the problem is coil whine in the power supply. 
 
The main "clue" is that when I change the power mode to "power saving", the buzzing mostly goes away.
 
So, I'm now back to trying to decide which PSU to get as a replacement for the Corsair DX850.
 
Since the Corsairs are notorious for coil whine, is there a particular brand that isn't (without costing too much)?

I'd swap out the power-supply.
If it were fan related, it wouldn't be "transmitted" thru the audio interface outputs.
If your power-supply is in the lower back section of the case, that's almost surely the culprit.
 
Ground issues are also common sources of noise.
 
I've seen poor quality USB cables introduce "digital hash" type noise.
 

I tried a different mouse, and connecting the mouse to different USB ports.  None of that made any difference.
 
Also, when the computer first starts up, it always lets off long SQUEEEEEALS, and this time I had my speakers on and hear the squealing really loud in them too. So, every time the back of the case makes noise, the speakers amplify it.

I found some balanced cables, and switched to those.  The noises went away.  So far, at least.  It could be a coincidence, though, since I didn't listen to unbalanced cabled immediately before switching to the balanced ones.
But at least there is some progress, even if it turns out to be short-lived.

To me it sounds like you are hearing two different noises and just associating them. Seems like a fan problem and a ground problem.

The power supply is about 6 years old.  Yesterday it (or something near it) was really noisy, squeaking like crazy most of the day.
 
I went ahead and ordered a new power supply (EVGA SuperNOVA G2 850W).  If that doesn't fix the problem, then maybe there's something wrong with the motherboard or the video card, neither of which I am eager to replace!

Yeah, I sure hope a new PSU is the silver bullet!  The high-pitched shrieking is really annoying, since I am on this computer all day!