In the television series Supernatural it is said that upon death, demons go back to hell and vampires and other creatures go to purgatory. Where do angels go when they die?
Actually it's never stated that demons go back to hell when they die. They do go back to hell when they're exorcised, and since yellow eyes was killed we haven't seen him except for flashbacks and such. Vampires and pretty much all other monsters go to purgatory when they die. It's never stated if angels go anywhere when they die, I believe they simply stop existing completely.
In season 8 we see Dean kill some vampires in purgatory, it's assumed that once you die in purgatory you stop existing.
This is similar to how we don't know where demons go, or where someone killed in Purgatory goes.
We know when they die they don't go back to heaven (which was obvious from the on-going war in heaven). We know that beings can only go to Purgatory if they have a soul. However, from The Man Who Would Be King:
CASTIEL You want to make a deal? With me? I'm an Angel, you ass. I don't have a soul to sell.
So, they can't go to Purgatory.
Therefore, at the of sounding a little wishy-washy, the angels go somewhere, and God has the ability to bring them back. Where that somewhere is, what the requirements are to get there, and how they get back, is a complete unknown.
As introduced in season 11: They both (angels and demon) go to the empty.
The Empty is a void that existed before God or the Darkness. It serves as an afterlife for angels and demons, where they sleep for eternity.
It's not clear yet. Misha thinks they're just gone but there must be some kind of loophole (See here). I think God must have some control over it, as he brought Cas back.
Well, demons can't stop existing because Death said that souls can be twisted tossed and turned, but never broken, not even by him. So souls can't be destroyed. Demons have souls, they maybe corrupted, but they are still souls. I think maybe they go to Purgatory when they die, because in Gods eyes they are probably monsters. And monsters have poisoned souls, Eve put a monstrous virus in their body and soul. And Leviathans have no souls, because they are older than souls (said so by death) so I like to think that purgatory is the place of the soulless and corrupted soul. God gave humans souls so that they can go somewhere after death. So to go to heaven or hell when you die, you need a pure soul. Now in this case pure doesn't mean good or righteous, it means not contaminated. Demons have corrupted souls, so they go to purgatory. God probably gave Angels grace, not only for power, but so they could go somewhere when they die besides purgatory. They don't have souls, so god gave them grace to take them to a special place for angels.I believe God made it so that when Angels take out their grace, or have it taken out, they gain as soul, so they go to heaven (or hell) not purgatory. So if Sam had died when he had no soul, he would have gone to purgatory.
Well actually, when Dean and Castiel went to Purgatory, they didn't actually die. It was just that when standing close enough to a Leviathan, you were sent to Purgatory along with it. So technically, Dean and Castiel were actually ALIVE in Purgatory, never actually dying. Also, if they did both die and go to Purgatory, it wouldn't work out, as Dean would either have to go to Heaven or Hell(being a human soul), and Castiel? Well, the answer for where he goes is still uncertain.