The only one I've seen so far that's claimed to be "Stable" is SlimKat:
http://xdaforums.com/showthread.php?t=2400577. Actually it looks like they just released a new stable version yesterday for our phone, Build 5.
I agree with this about slim. It's a very stable ROM. There are however, alternatives. I'd suggest you try a few and see which ROM's have the best combination of features and stability for you. I've been a huge fan of slim for quite some time now, but recently, I've been running Nameless ROM. I've ran a nightly from the first of the month, and since then have had only one instance where the phone randomly rebooted. In fact, that's the only time I've rebooted it. LOL. For me it has the features I'm looking for without having to resort to using the Xposed framework. I have nothing against Xposed, it's always worked great for me, but I just prefer to have features baked in if at all possible.
I'm sure there's a ton of other good roms too, many people like Gummy, Beanstalk, Paranoid, PACMAN (can't think of the exact name), and GideonX's builds of various roms (check them out at droidhype.net), liquid, Carbon, and other's are nice too. Many have too many featues for me, and some have not enough or lack what I'm looking for.
So again, Try a few and see what YOU like. That's the best answer I can give. I doubt that many are "stable", but they've been stable enough for me. As I mentioned with Nameless, they're not labeled as stable, and nightlies are just that, nightlies, but you'll find that most nightlies are "stable"
