TerraSin
I can't speak highly enough for the Corsair Obsidian Series 550D case. Everything about it is based around keeping it silent.
http://www.newegg.com/Product/Product.aspx?Item=N82E16811139012
 
For fans, I use SilenX. They work very well at keeping the case cool while getting good airflow.
http://www.newegg.com/Product/Product.aspx?Item=N82E16835226042
 
The new desk I'm building will have an enclosed area for the computer case with two fans on the back to promote airflow through it while keeping sound inside it from escaping as much as possible.

Does that case include a power supply?  It didn't seem to.  what power supply are you running with it?  Also, did you replace the fans that came with the case or did you add the silenX?
Thank you

Anyone have any experience with this?
http://www.microcenter.co...X_Modular_Power_Supply

GPU fans are the biggest source of noise, along with noisy chassis fans. 120mm chassis fans have a higher CFM rate at a lower speed i.e. they can dissipate hot air by shifting more air with less rotation. If you are serious about silence I would look into getting a corsair case and a gpu with passive heatsink.

My DAW is very quiet indeed - measuring under 21dB at 1 metre. I actually can't get an accurate reading because neither my cheap sound meter or the very expensive UK-government issue one my wife has access to reliably read that low. Any noise picked up by mics pretty much disappears into the general noise floor.

Cpu is cooled by a big Noctua heatsink with a 140mm Noctua fan I can switch to run at 5 volts (PC standard fan voltage is 12 volts).
Fanless Seasonic PSU.
Fanless Sapphire-branded AMD gpu.
Two very quiet HDDS, two silent SSDs.
The cpu isn't particularly power-hungry, the 6 core upwards ones with a disipation of 140 watts would be a much tougher challenge to cool as quietly.
The only fan besides the cpu is another Noctua 140mm also switchable to 5 volts as a case exhaust fan.

I usually run the fans at around 7.5volts, but when I need near silence I drop them to 5. Having them permanently at 5 isn't good because that's approaching the point where the fan won't have enough power to start up.

The case makes surprisingly little difference to noise. Noise will escape the PC case if there is a hole in it and there have to be holes. What does help is a solidly built case with deadening material on the panels to help stop the case itself from resonating in sympathy with the fan noise.

Building a quiet PC can be done, but it takes careful selection of components and a willingness to experiment and test. Before running fans at low voltage for example, stress test the cpu and gpu and check their temperatures. Lower fan voltage and repeat. Tests need to be several minutes duration with components at 100% or as close as possible. A willingness to run cpus and gpus at higher than 'normal' temperatures (but still well within their rated thermal range) helps.

Then try shifting the case fan around, say from the front blowing in to the back blowing out and see if it makes a difference. Vents in the case top to let out hot air can help.

www.silentpcreview.com is a useful site, as is the already mentioned Quiet PC.

Oh, finally, regard manufacturer's claims about 'low noise' products with a bit of scepticism, unless that manufacturer is Noctua or Scythe. Zalman are best avoided altogether.

I can HIGHLY recommend this case: http://www.coolermaster.com/case/mid-tower/silencio-550/ 
 
It's the one I use in my control room and it's dead quiet. Also really cool is the sideways-fitted plastic pressure-fit HDD clamps so you can pop drives in and out in seconds.

when i built my own DAW, i used this for a case:
 
Antec Solo Black/Silver Steel ATX Mid Tower Computer Case
 
and power supply:
Thermaltake TR2 RX W0134RU 550W ATX12V Ver2.2 SLI Ready Modular Passive PFC PFC Power Supply
 
 
very quiet, without going overboard.
 

I like tlw's advice here ... if you choose quiet internal components to begin with, the case becomes less of a factor.  Key elements:  big CPU cooler with quiet fan (e.g. Noctua), fanless PSU / GPU, SSDs or 2.5" HDDs in vibration-reducing 3.5" adapters (e.g. NoVibes), quiet case fans.    A PC can be practically silent even with the case wide open.
 
Cheers,
Eddie
 

If you want the most silent power supply, go with a fanless one. There are a number out there you can find and so long as you aren't doing high end gaming, they'll all be more than fine for your computer. As ever, I'm a Seasonic fanboy so the Seasonic Platinum Series Fanless is what I'd look at. No fan noise, only some very minor electronics noise (all electronics make a bit of noise just in operation).

 
For fans the key is large and slow spinning. If you keep the system with just the CPU in it (no discrete graphics card) one case fan should be fine. I'd look at Noctua and Phanteks. The Noctua NF-P14 FLX is my top pick, so long as it is mounted on the side of the case, not a top (they have trouble spinning in that alignment). Their noise is low and the profile is unoffensive, and the static pressure they have is quite good. For CPU cooling, you want to get something with a nice large fan on it again so it can spin slow. Arctic Cooling is my go-to company for CPU coolers. A Freezer 13 is probably what I'd look at. Very overkill, but because of that the fan should rarely, if ever, spin above minimum.  A Freezer i11 would work fine too.
 
You can get cases that do purely passive cooling, but I think it is more effort and expense than it is worth. Good fans are quiet enough.
 
Next issue is harddrives. Those suckers make more noise than you might think. If you have the money and can go all SSD, problem solved. If not, then you want to try and keep them dampened, if possible. I'd stay away from WD drives and go with Seagate or HGST, they make less noise in my experience (even though HGST and WD are the same parent company).
 
A case can of course help keep all the noise in. My top quiet case is an Antec P183B. They do a really good job with isolation in that case, building it out of multiple layers in a metal-plastic-metal configuration to attenuate noise, and they have dampening on everything including some nice silicone grommets for the HDDs.
 
If you want to attenuate it further, you can get acoustic foam for the inside of your case. Putting it on the walls will help stop the higher frequency fan noise from going anywhere and make it a little quieter.
 
Also make sure to consider your monitor. If you get a really quiet room, monitors can become an audible noise source in some cases. I love my PA301W, but it buzzes audibly. It isn't loud, but you can hear it when the room is silent since the ambient noise level is down around 20dB. TFTCentral does a good job reporting on monitors making noise so check on that before you get one. In general, LED monitors that do not use PWM dimming tend to make less noise than CCFL monitors or monitors that use PWM.