Both stacks and stratifolds can model spaces with singularities (say, singular quotient space for example). In my little experience, stacks are widely used in moduli problems and stratifolds are often used in singular symplectic reduction (see Introduction to Differentiable Stacks (and gerbes, moduli spaces ...) and A Survey of Singular Reduction for quick introductions).
They do have some ideas in common: they are constructed to "remember" the stabilizers somehow. My question is, what are their (respective) pros and cons in modelling spaces with singularities? Or, when do we use stacks and when do we use stratifolds in general? Are they "equivalent" in some sense?
