For the Bugs and stability report head towards XDA thread.Before stating make sure you have custom recovery in your device and make Backup of current rom before flashing CM13 in galaxy S3.So lest start the step by step guide on how to install CM13 rom in Samsung galaxy S3 AT&T
1.Your device must be rooted
2.Any custom recovery is required (TWRP/CWM).
3.Make backup of current Rom before flashing Cm13 rom.
#How To Install CM13 rom In Samsung Galaxy S3
1. First make sure you have downloaded all the necessary files from the Downloads section below.
2. Place the .zip files in your phones internal or external storage from where you could access it eaisly.
3. Now boot into TWRP recovery,
(Optional – Backup all your partitions if you want, so that if anything
goes wrong you could restore your data back, its a lifesaver), then
perform a factory reset.
4. Now install the ROM files by
tapping on install from TWRP’s home screen, then locate the .zip files
you placed earlier and swipe to flash, (Remember – First flash the ROM
then Gapps package).
5. After its done, reboot your device.