The only shaving sticks which are a bit of bother are Arko/Derby. They have paper only, and I rip off most of it, but leave a little at the bottom to hold, I've wrapped the end in a bit of kitchen foil so the paper doesn't stick to my fingers. After use I put it into an old box from a shaving brush or something, then put away.
The ones which have a dedicated container, eg La Toja are the easiest to store.
The ones which have a plastic base holder eg Palmolive, WIlkinson Sword, I hold the plastic bit when I'm using it in the shave, then put it on top of the bathroom cupboard to dry after use, then when it is dry I put it back into the box then put away into my cuboard.
When I have travelled I used a shaving stick that had it's own container.
I find shaving sticks to be the easiest to store, but I do not travel for work like some eg Mike Smart above.