They're always supposed to be upright ... in use they otherwise can feed liquid propane instead of gas to your appliances (unless as noted above, they are e.g. lift truck tanks designed to operate horizontally). Don't know whether your tank has an overpressure vent, but if so and on side in storage, if it should overheat / overpressure say on a hot day and vent a bit, it might vent liquid propane - a much worse scenario than a little gas "burp"
At my local propane dealer, they have photos of an auto in which two women were allegedly transporting a newly-filled cylinder a) on its side and b) in the trunk. Purportedly it did leak and the vaporized propane ignited and both were burned to death. The car is hardly recognizable as a car after the fire. They absolutely refuse to fill any cylinder unless it is obvious that it will be taken from their premises vertical, restrained from falling over, and not in the interior space of a passenger vehicle.
I once witnessed a BLEVE (boiling liquid expanding vapor explosion) of a propane tank that was on it's side, leaked, the gas ignited and stayed lit, heating the tank until it ruptured. It's an event not to be experienced up close, and usually involve a tank on it's side developing a leak. Here's an example of a big one:
YouTube - Train car explosion bleve
Anyway, I always store mine vertical.