Inside. Too easy to bump the outside against the tree or other obstacle and crush your saw.
Wiley P, congrats on the bucket truck! You are now required to post pictures immediately in the OCGD thread.
Squishey, do yourself a favor and buy or build a scabbard. The fiberglass ones are nice but expensive, and you need the ones with the wear pads built in or else they get chewed up too fast. Or you can make one out of plywood, and Andrew has shown how to make one out of 4"-6" pvc pipe.
Do your boots get nicked up badly?I voted inside as I don't have a scabbard so the outside isn't even a option.