There are a lot of popular culture references in the show. In the first season especially the monster of the week is often cliche monsters and there seems to be in jokes and parodies of other horror films/tv shows that they reference. Also the later seasons of the show deal with the idea of God and heaven and is sometimes ambiguous. The whole idea of the supernatural existing is something that has been quite a dominant argument from modernism to postmodernism.

So would you say Supernatural has a strong enough case to be postmodern, if so for what main reasons?

I would say that it is modern. It is meant to be a depiction of what Sam and Dean's world (and the world in general) would be like in the present (ours as well as theirs), if these supernatural things existed. That would be the reason for the pop culture references as well. They enforce the reality of the show.
The idea of the supernatural can be either or, but the show itself is not postmodern.

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