The Falcon 9 stages, by themselves, are usually stored on rotisserie style mounts, that allow the stage to be rotated and worked on easily. That is, if the engineer needs access to the top, instead of bringing a ladder or scaffolding over, and possibly dropping something on the stage, they roll the stage till the components are at the bottom.
Here is the mount, on a the Orbcomm stage, in the factory, with the engine install rig (The double ring of metal) being pulled back. Look just north of the engines, there is an aluminum ring around the stage.
Here you can see three recovered stages with the ring in site and in various states of engine removal.
Here is another recovered stage, where you can see the rings at both ends.
Here is a shot where you can see the ring from above.
You can see the huge amount of scaffolding they had to deploy for the Shuttle to work in the payload bay, leading to much slower turn around times.
The Saturn V had custom scaffolding in the VAB as well.
With the Falcon Heavy, at some point they have to attach the three cores. Do they attach the cores on the rotisserie stands, or do they wait, bring in the TEL and assemble them together on the TEL itself?