I was shocked to be unable to find any systematic reviews on the relevance of Jung and particulary his archetypes. In my search, I usually encountered criticism based on definitions too loose to be properly debunked but I think the concept of archetypes as "distinct universal personalities in our brain" is quite a falsifiable one.
I find Tom Boardsman's SE answer to a theoretical question on the archetypes really interesting and I think it does provide an argument against the archetypes as Jung described them - as literal images - from what I've gathered, automatic visual detection is pretty rare in humans and can be found only in a few areas such as face recognition or snake detection. Jung's idea of such images made a lot more sense to me when I found out it was likely influenced by lamarckism. But - of course - my layman notion is far from a proof.
However, I imagine Jung's archetypes could be best put to a test by a cross-cultural comparison of dreams. Sadly, I haven't found any study that would take this challenge seriously. A bit less exact way could be cross-cultural comparison of narratives for behavioral patterns. In that regard, I have only found mentions of criticism of the remarks on the very non-Western cultures that influenced Jung.
So are there any studies which would provide some basic evidence for or against the existence of Jungian archetypes?