LP boxed sets - best storage

As others have said, store your LPs vertically.  I have a lot of box sets (100s), many from the 60's and 70's, and have always stored them vertically.

Boxed sets used to come with full-size foam sheets inside. When new they had some resilience. Their purpose was to fill the empty space and put some side pressure on the LPs, to prevent warping, as we do by storing our single LPs vertically side by side. The foam was even used in different thicknesses, depending on the number of LPs and size of the box. That foam lost its spring then rotted long ago, leaving sticky crumbly bits that stick to both sleeves and vinyl — a total PITA. But they once served a purpose and did so effectively. If you're concerned about loose LPs warping in a box, a thin square of styrofoam may serve the same purpose. They say styrofoam cups last 20,000 years in a landfill, so I doubt it will rot away in your collection of Bach Cantatas, as the old foam did. Thin bubble-wrap also works, as do corrugated cardboard squares, and you can build the thickness by adding layers, but it can get a bit OCD. It's a cheap and easy remedy if warping is a concern to you. It isn't to me. 

In a related matter, I see many LP collections in which part of each storage compartment is empty, the LP's in each leaning heavily to one side. Not good! What I've done for years is use the 12 X 12 pieces of cardboard packing used in LP shipping cartons to fill the empty space in each compartment of my LP racks. That keeps the LP's firmly supported vertically. Local LP sellers will happily give you all they have, just ask.

I use cigar boxes, which don't fill the entire gap, but keep the LPs pretty straight vertically. Looks better than cardboard. But you have to shuffle things around as your collection grows.