This is a bit of a grey area.
The majority of phones are not rooted by default. Rooting, falls into the category of hacking. However discussing the use of a rooted phone, or open source software such as Android is entirely allowed.
As long as this thread doesn't turn into a how to for rooting, and no questions are asked in regards to methods of how to do so - the thread can remain open.
That being said, I personally use CyanogenMod - I find they have excellent dev's and the ROM packs a lot of useful features you'd normally have to get an app for. Like CPU controllers to allow setting min/max clock speeds without having to install a third party app from Play Store.
MIUI is good too, although I don't know whether or not there's a ROM for the S3 (I have an HTC Desire Z). Also, if you try MIUI - download the ROM from either the American, United Kingdom, or Australian mirrors. The ROM on the official mirror is in Chinese.