Sandpaper is such a key finishing tool that I wanted to make it handy to my lathe. So, in building my tool storage rack (located behind me when I stand at my lathe), I included a section for my sandpaper. The section is divided into 2 areas; storage of new sandpaper and storage of partially used sandpaper.
(click on any image to get a larger view)

Here is the big view of the sandpaper storage area. It consists of the clothespin rack and the new paper storage shelf. See below for more detail of each area.
The new sandpaper shelf hold news paper that has been torn into eights on a straight edge (normally the side of my table saw). Each grit of paper has it's own storage area marked by the grit size. The entire shelf is covered by a piece of hinged plexiglass. You can see the hinge in this picture but can't see the plexiglass as it's clear.
The partially used sandpaper storage area is a piece of wire through a set of clothespins. The clothespins have the grit size marked on them. I resist the urge to reuse sandpaper (since it's no longer sharp and is a huge waste of time) but often find that I've used only one or two sides of these 3 sided sheets of sandpaper.