Many algebraic geometers have encountered some notion of semistability, either in studying a moduli problem or in the more concrete context of geometric invariant theory. Often along with a notion of a semistable point, one can stratify the set of unstable points by some numerical quantity encoding the degree of instability of each point. I will discuss some of the very special properties of the resulting stratifications -- both topological and homological -- and how they can be used to deduce information about the semistable locus itself. I will also discuss a new framework for studying such stratifications that generalizes the constructions in geometric invariant theory and well-known modular examples.