I was running Cyanogenmod 11 (android 4.4.4) on my Samsung S3, using a SNAPSHOT from Aug, 2015 and not getting any more updates. Finally looking into why it turns out they had moved on to developing for Lolipop (5.x) so to upgrade I'd need to wipe and install from recovery. I decided to go with a Nightly build this time because life is short. Besides, recently I noticed that mobile data (LTE) no longer worked and maybe an upgrade would fix it. I usually only use wifi, but when I'm out and about and need connection someplace I'd like it to work! Also, having recently gone to 5.x (android lollipop) on my tablets I thought it was time to upgrade the S3 as well!
My S3's home button stopped working long ago which was initially a pain but after enabling the on screen buttons and using the device like that for some time it no longer bothered me. However, I didn't realize until I wanted to upgrade that having no home button makes it impossible to get into recovery using the manual button method (pressing Home+Volume down+Power). I realized that I was still able to get into recovery (which in my case is Clockworkmod recovery 220.127.116.11) by connecting up to the laptop with USB, turning on USB debugging and issuing
adb reboot recovery in a terminal (I run Linux on my laptop and already had adb all set up from working on my ASUS Transformer). Luckily Clockworkmod can be controlled through gestures (swiping up and down and right to select) so the home button wasn't necessary there either. Then I chose to wipe the device (factory reset), and flashed the most recent NIGHTLY version of CM12.1 just to live extra dangerously and an appropriate gapps zip (which I had copied over while connected up over USB) and rebooted.
The setting up phase for Lolipop had begun, and since I had chosen to back up my apps to Google Play on this device they started to install again straightaway. Having used it for a few days now I can say that it feels faster and pretty much everything works... except for the mobile data. I'm using Ting (which saves me massive money on cell service - I don't think I'd have a cell phone at all if it wasn't for Ting) and I messed around with APN settings and since the S3 is rooted, files in /system but alas, I can't seem to get it online. Someone said that its been messed up since last April but for now I think I am just going to live with it and keep an eye on the change log for the Nightlies for a fix.
List of Cyanogenmod roms for the Sprint S3 http://get.cm/?device=d2spr
Changelog for D2SPR http://www.cmxlog.com/12.1/d2spr/
Clockworkmod Recovery http://forum.xda-developers.com/wiki/ClockworkMod_Recovery
The versions I used successfully:
cm-12.1-20160216-NIGHTLY-d2spr.zip (279.7 MB)
Gapps (Google Apps)
open_gapps-arm-5.1-stock-20160215.zip (541.7 MB)