Depends on which carrier/plates and length of storage. I've made a few stands, like the one from ITS, but that's better for the storage of something heavy like a kitted-up IOTV or similar. It makes for a cool "decoration" in a man cave or something, but it's a lot for just a plate carrier that's going to be deployed rapidly IMO.
I personally store my plates/armor separately just to reduce strain on the rack in my closet and I have more carriers than plates/armor. I personally don't need to have several fully kitted-up PC's just hanging in a closet. If I decide to wear/do something different than what I'm currently using, I have the time to configure it as needed. I store soft armor in a stack in a closet, same goes with plates.
Right now, I'm using a US Palm Defender with LTC 26225's with the Armor Express IIIA inserts for HD duty. It's way over-loaded and you wouldn't want to wear it for more than 60-90 seconds
but it's stupid fast/simple to put on in a hurry. I'm sure there's better options but I've seen a few other guys run this setup and it works pretty well for around the house. I lay this one flat on the carpet and have no worries about it... I would like to store it vertically to save space but I notice that if I do that the soft armor can shift and the plate could cause it to "bunch up". Yeah, I know the plates are level IV but having the inserts in there makes it slightly less miserable to wear... If I was to store it vertically, I would take care to make sure it didn't fall over.
I prefer storing flat as I think I'm more likely to drop my PC off a hanger while trying to put it on and inadvertently break my foot.