I cant see any harm in storing small propane cylinders on theire sides.
The contents are liquid, with an expansion space containing gas/vapour.
The cylinders must however be used the right way up, with the valve or connection at the top. If a cylinder was put to use the wrong way up, liquid rather than gas could enter the burner and cause it to flare up, with risk of fire or damage to the appliance.

I would advise against storing propane indoors in a home or attached garage.
As pointed out above the containers sometimes leak, and this leakage is liable to pool in low areas, any spark from a furnace, sump pump, refrigerator etc could then cause a disasterous explosion.

If circumstances make indoor storeage unavoidable then use only well known brands, and keep as far away as possible from ignition sources. Consider buying a leak detector, these are available from marine suppliers and are intended to detect any collection of gas in boat bilges.