All right, so this part of the thread will be a listing of custom ROM's which I can confirm as working on our device, and the necessary links. It won't necessarily be a true one-stop shop, since it will be almost exclusively 4.3.x builds (and now 4.4.x since KitKat is here!), but that is just my preference as mentioned earlier!
Side Note: Since I tend to hoard files on my computer for every device I have ever owned, chances are I have files for some of these ROM's that are going to be hard to find, especially those looking for 4.3.x builds for ROM's that are now releasing 4.4.x builds. If this is the case, please send me a PM and I will see what I can do for you. Keep in mind, these will all have been modified to my personal tastes-see post #2 for the .apk files I remove from ROM .zip files prior to flashing, in addition to removal of the assert lines.
CyanogenMod
The granddaddy of them all. Without this one, pretty much none of the rest would be what they are today. Main site:
CyanogenMod, XDA discussion thread:
XDA Thread for CM 10.2 and CM 11. As this one is pretty much the "pioneer" pay close attention to what other users have experienced with the various nightlies; some are stable while others have bugs which would make certain nightlies not worth flashing.
Update 12/22/2013:
I have been running official CM 11 very heavily since it was released on 12/7/2013, and it is one of the most awesome set of nightlies I have ever experienced! Very stable, any bugs are being addressed very quickly by the CM team. I have a thread on both the VM and the Boost forums dedicated to CyanogenMod, they include Dropbox links to specific nightlies which I have modified, and in some cases debloated, for ease of use.
Virgin Mobile All Things Root Forum Thread for CM 11 Discussion
Boost Mobile All Things Root Forum Thread for CM 11 Discussion
Both threads are updated by myself with the same material in the OP, as needed, but there are unique discussions going on in both threads. Thanks as always to my new friends on the Boost side who are making VM and Boost a united force!
Carbon
Hands down my favorite ROM. Extremely smooth and stable, awesome customizations. I especially like the "Ribbons" feature, which creates a dock which can be swiped from various areas of the screen but otherwise remains hidden. I disable the dock in Nova Launcher to maximize my screen real estate, so Ribbons makes it convenient to access frequently used apps. Listview animations are also pretty interesting, these add animation features when scrolling through any list such as the Settings menu. For those that like it, HALO is also available. Download page here:
Download Carbon ROM, make sure to choose d2spr folder; XDA thread here:
XDA Thread for Carbon ROM.
Update 12/22/2013:
Carbon now has their official website up and running, with a nice overview of the features available as well as a promise that their 4.4.x builds are coming any day now. Unfortunately, that tease was put out a while ago, with the disclaimer that the Carbon team wants an awesome release, not something done halfway which will be a work in progress. I am assuming it will therefore be awesome pretty much right off the bat, if 4.3.1 is any indication! Maybe a Christmas release for their diehard users? We will know in a couple of days! Once Carbon does start releasing 4.4.x builds, I will start a dedicated thread similar to my threads for CM 11.
Official Carbon ROM Website
Hellfire
Similar to Carbon in terms of customization options, with a red color theme and the TouchWiz water drop sound when screen touch sounds are enabled. As with any CM based ROM, custom themes are available for download from Play Store if you don't like the default theme that comes with any of these ROM's.
XDA Thread for HellFire ROM, with download links in the OP
"The Collective" AoCP
Decent ROM with some twists on theming built into the default ROM theme; very annoying screen touch sounds however, in my opinion.
XDA Thread for AoCP "The Collective" with download links in OP
Update 12/22/2013:
My buddy wetbiker7 from the Boost forums strongly suggests not using this ROM due to instability issues that develop over time; Biker is pretty much the man, so if he suggests something, you can bet it will end up becoming my suggestion as well. Leaving the ROM in the list just for the sake of being "complete," but would suggest trying it only if you are really curious. Doesn't really offer anything above and beyond any similar ROM, such as Carbon, or CyanogenMod with the proper Xposed modules.
JellyBAM
Probably my second favorite after Carbon. However, apparently some kind of issue where they didn't give proper credit to the original ROM developers from whose source they compiled (this ROM is kind of like PAC-Man in that it borrows the "best of" from multiple other ROM's), which resulted in them getting banned from XDA. Download from Goo.im here:
JellyBAM Downloads; they just released a version which I believe correlates to a stable release for 4.3.1. GApps are included in the ROM, but if you want very current GApps, use the ones provided by Paranoid Android.
Update 12/22/2013:
JellyBAM is also now on the KitKat bandwagon, they are using 2.x as their numbering system (1.x was for their JellyBean 4.3.x builds); however, currently reported as being fairly buggy, so it may be a while before the 2.x builds are as good as the 1.x builds. Check out their official website:
JellyBAM
PAC-Man
As mentioned above, kind of a "best of" (they call it "All-in-One") featuring Paranoid Android (P), AOKP (A) and CyanogenMod (C), hence the name. This is a well-respected ROM across a lot of different platforms, just be careful with some of their latest nightlies (XDA is a good resource to check out what issues other users are having with certain releases, for any given ROM).
XDA Thread for PAC-Man ROM, download links in OP
LiquidSmooth
Pretty nice ROM, but hasn't been updated in a while (over a month as of this writing). If you combine this with a custom kernel, pay close attention to build dates for compatibility.
XDA Thread for LiquidSmooth ROM, download links in OP
Update 11/10/2013:
LiquidSmooth v. 2.37 is now available; clean flash went fine except could not get a working data connection or any phone signal whatsoever, restored my backup of v. 2.10, then dirty flashed v. 2.37 and data/signal were OK; most likely my problems were due to coming straight from Omni, and yes, I did all the correct procedures for wiping partitions, cache, and Dalvik-cache first.
Update 12/22/2013:
LiquidSmooth is just now starting to release builds based on KitKat. Will update their section of this thread once the builds are truly up and running.
Illusion
Also pretty nice, but similar to LiquidSmooth in that it hasn't received a fresh build in a while (again, as of this writing).
XDA Thread for Illusion ROM, download links in OP
The following ROM's I only flashed just to confirm working, otherwise I have not actually left them running on my phone long enough to form an opinion.
Omni
I am only going to mention Omni, no links or recommendations. When it was first ported to the SGS3, there were fairly severe problems with the ROM, and it seems like, a couple of months after that initial port, that there just really isn't much going on for our specific phone. While I am impressed by the concept behind Omni, I was less than impressed with what it actually had to offer for the brief time I had it running. If you want to check it out, Google will be your friend.
ReVolt
XDA Thread for ReVolt ROM
Eclipse
XDA Thread for Eclipse ROM
AOKP
XDA Thread for AOKP ROM
CyanFox
XDA Thread for CyanFox
Update 12/22/2013:
While I was not that impressed with the 4.3.x builds for this ROM, 4.4.x has been a different story. Very solid, and combining this ROM with the proper Xposed modules brings it on par with Cyanogen or Carbon in my book. Prior to official CM 11 nightlies being released, this was my ROM of choice for the KitKat experience. Once Carbon starts releasing their KitKat builds, CyanFox will be a top three ROM for me, with Cyanogen rounding out the "holy trinity" of KitKat-based ROM's in my book.
Vanir
XDA Thread for Vanir ROM
Despite the disclaimer for this section about these just being confirmed as able to flash, I did form an opinion in the five minutes I had it installed-pretty much worthless! Way too many bugs, including a non-functioning home button. Only remains on the list to be "complete," flash at your own risk.
Slim
This is the only one I haven't tested at all, but am pretty sure it is capable of being modified and flashed. Used this on other phones, didn't care for the DPI changes, and it always seemed to get laggy over time, but again, I don't know how it performs on the SGS3.
XDA Thread for Slim ROM (currently SlimKat)
Update 12/22/2013:
My buddy
wetbiker7 really likes this ROM and says it runs really well for him. I trust his opinion, so even though I will probably never actually run this ROM, if he says it's golden, then you can flash with confidence!
ROM's That Have Become Available Since Release of KitKat
Here are several ROM's that have been ported or built from scratch since the KitKat update to the Android ecosystem:
Beanstalk
Confirmed to run well by
wetbiker7, so good enough for me!
XDA Thread for Beanstalk
Gummy
XDA Thread for Gummy ROM
Smasher X,
Epic 8, and
iDroid
A trio of ROM's from talented developer
Cjen1987 over on XDA. I have only personally tried Smasher X, it was fairly slick with lots of customization and a unique home launcher included (Smart Launcher, available on the Play Store). Developer updates very frequently, and is open to user input.
XDA Thread for Smasher X, Epic 8, and iDroid
Honorable mention (because it is a 4.2 build):
Team BAKED BlackBean 8; I love this ROM on my Motorola Xoom, and it does flash on the SGS3. There are no assert lines included (Illusion also has no assert lines), so no modification needed in that area, as always just make sure it is for d2spr. Goo.im link here:
Team BAKED
I cannot stress enough, that no matter which ROM (or ROM's) you choose to play with, read, read, read and then read some more on the various threads to get familiar with the pros and cons and to see what other people have already experienced.
Happy Flashing!