I use "Camera ICS", which I downloaded from the Play store for free. It seemed the easiest and safest way to fix this problem (shutter sound.) It is a stand-alone camera app, not some sort of patch for your stock camera app, so once you download the app, you open it instead of your stock app and it will have an option in the settings to silence the shutter sound. Still takes high-quality photos because it is still using the same lens and what-not (of course), but I haven't checked to see if it has all the fancy options that come with the stock app...but for simple and silent photos, it is definitely good.
I am on the Samsung Galaxy SIII, and Sprint is my carrier here in the US (Florida). I previously had the GSII (also with Sprint) and it had the same problem. Glad this app is out there, because that obnoxious shutter sounds was way too loud.
I see a lot of people answering by saying "it is there, you just have to look harder"...perhaps there are a few people out there who are completely inept with their phones, but most of us know our way around our gadgets - at least well enough to fully search the settings within the camera app, so "look harder" is not a solution. If you are on a different carrier, in a different country, or perhaps even a different region within the same country, that which is true for your phone may not be true for other phones. My phone doesn't have the "silent shutter" option within the settings, and muting the phone completely with the drop-down menu is equally ineffective at silencing the shutter.
I don't want to root my phone until I feel it is necessary for better performance, so this app seemed the easiest way to go (someone else actually recommended it on this forum).
Also, to those who are (half) jokingly inquiring as to the reason for wanting to silence the shutter, there are innocent reasons for wanting this. Just a couple weeks ago, I was doing a community service thingy for underprivileged kids at a childcare facility here in Tampa (volunteer stuff, not like a probation deal or anything, lol), and they were all taking turns opening gifts and sitting on Santa's lap. Having this loud shutter sound go off every few seconds while kids are whispering into Santa's ear about what they want for Christmas is extremely distracting and quite rude. So while it may be near certainty that "perving" is mankind's oldest and greatest past-time, there are other reasons for wanting to silence this completely unnecessary sound effect.