I don't know about 360 video but the software video stabilizers are getting pretty good. I've used a version of Mercalli that is built into the Corel VideoStudio software that has done a really nice job of stabilizing shaky video. The most impressive one I've done is where my camera was resting on the deck of a paddle board facing forward while I paddled. The software processing really did the trick and the horizon was rock steady while you could see the bow of the board rocking back and forth wildly (when in reality it was the other way around). It's pretty cool stuff.
The embedded Mercalli software in that VideoStudio package is somewhat limited because of operating system limitations. Apparently you can get a straight up package from Mercalli that does a little more.
If you're a Mac user you could get a copy of Apple Motion from the appstore ($50) which also supports advanced rotation and motion tracking.
Not sure how 360 video would work with rotation though since the unprocessed footage is very different than standard 2D footage.
