How do you store and catalog your vinyl?

I have about 5,000 LPs.

They are stored on shelving that consists of 42 cubes, 15 x 15. I built them myself.

They are divided into into three more or less equal sized categories: classical, Afro Cuban, and everything else.

The classical is divided into early (pre-baroque), classical, modern (Alban Berg, Schoenberg etc and after), then arranged alphabetically by composer, then by date composed.

Afro Cuban is arranged alphabetically by artist, then by date of release.

Everything else is divided into jazz, blues, gospel, folk, world, reggae, Afro-pop, country and 50s-80s rock n’ roll, rock, then arranged alphabetically by artist, then by date of release.

Some LPs can fall into one or more categories, I make a call and hope I remember. Eg aretha is both rock and gospel.

The 200 78s are arranged alphabetically.

As for inventory, I am in the process of photographing each LP cover, front and back, on my iPhone. I’ll download them into a document that I will notate regarding condition etc.

i tend to listen to the same 700 or so regularly. About 3,000 get listened to less regularly. About 500 are there because a guest will ask to hear it. About 700 need to go.

bdp24: I file Dylan and Cooder under folk. 

My previous reply was maybe a bit fallacious.  Truthfully, I divide my albums into "classical", "jazz", "rock" and "folk" music.  I also have a "new album" section, and a "current interest" section, which are somewhat intermingled.  I have a small stack that is in current rotation.  I have about 400 albums, and never have any problem finding what I am looking for.

Ikea Kallax is a Godsend. I have many of them to house my 4,000+ vinyl, organized alphabetically, with separate sections for rock/pop, jazz, audiophile and classical. I have never seen the point in cataloguing them, except that it might prevent acquiring more than one copy by mistake.....

Agreed on Ikea, I hauled 5 units up from Orlando in my old El Camino, poor thing was dragging its behind with the weight....lol.
Of course couple months after that they opened an Ikea in my town, go figure!