I have a set of almost new rims with almost new tires. The are mounted and balanced and are not on the vehical. Should they be stored inflated or deflated for best shelf life.

blau

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Silver Honda Accord, 2006
Silver G500, 2003
Silver SLK-320, 2002
Black ML-320, 2000
Bule Porsche 993 Targa , 1997
Silver Merkur XR4Ti, 1987

Best stored inflated to maintain the seal between the tire bead and the lip of the rim.

Also, store the wheel/tire laying down on a flat surface (dirty side down, so as to keep the shiny side undamaged). This also inhibits the formation of "flat" spots on the tire, which is common on tires still mounted on vehicles stored for long periods of time.

The "flat" spot issue is more of a problem with older bias-ply tires, not the newer radial compounds...

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2009 ML350 (106K) - Family vehicle
2001 CLK430 Cabriolet (80K) - Wife's car
2005 BMW 645CI (138K) - My daily driver
2016 Mustang (32K) - Daughter's car

Would it be recommended that you store them on hooks? (against the wall?)
Just curious as I do not have any spare room in my garage floor, and I am contemplating buying two sets of extra rims for winter tires. That would take up some serious space!

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Current: '91 300TE 4MATIC 317k and climbing...
Former:
'81 300TD Wagon 168K "Tank"
'83 240D 216K 4spd manual "Da Bear" (aka best car ever)

"Never sweat the petty things...
and never pet the sweaty things."

I don't see a problem with mounting the wheels/tires on the wall to save space.

The "flat spot" issue is not due to orientation, but the constant vehicle load on the same tire surface area for extended periods of time.

__________________
2009 ML350 (106K) - Family vehicle
2001 CLK430 Cabriolet (80K) - Wife's car
2005 BMW 645CI (138K) - My daily driver
2016 Mustang (32K) - Daughter's car