I store mine vertically in Cropper Hoppers (see below):
Close up of Jetmax By Siameseplease
They are now labeled. I sort them by manufacturer, for the most part. I do have cat themed, military/US, and Christmas themed in their own bins, though.
For cardstock, I use a paper rack like Jen's and have it sorted by color.
I also happen to have a stack of various PP on the floor right now, that I haven't yet sorted into appropriate containers. I just keep stealing papers out of it to work on my cards.