You could pretty much use any of the forms of storage that you use for other types of paper storage.
The thinner ones, I store in my file folders in my scrapbook paper cart. (Thicker ones don't work as well this way of course - although once I've used a bunch of papers this becomes a better option for keeping the remaining pieces all together for future projects.)
The thicker ones, I just store upright on my bookcase and prop up with book ends if need be.
But you could do as Aquabunny does or even use the vertical magazine holders with labeled dividers if you think it helps.
Or you can put them in Sterlite type drawers or Iris containers...
You could use oversized baskets and stand them up in them.
The list goes on and on. It is pretty much personal choice. Hope you find the idea that fits your personality and budget.