If I were to have asked this question about local systems, it would be extremely easy to give examples from physics/biology, etc. I can give example if people want, but this should be easy to Google.
Question: What examples of perverse sheaves naturally show up in nature?
More specifically, I mean examples which are not local systems, and the example only applies to perverse sheaves (or constructible complexes of sheaves) and not sheaves of vector spaces in general.