A central problem in local dynamics is the equivalence problem: when are two systems locally equivalent under a change of coordinates? In the neighborhood of a singular point, representatives of equivalence classes could be given by normal forms. But, most often, the changes of coordinates to normal form diverge. Why? What does it mean? In this talk, we will discuss a class of singularities for which we can provide moduli spaces for the equivalence problem. We will explain the common geometric features of these singularities, and how the study of the unfolding of these singularities allows understanding both the singularities themselves, and the obstructions to the existence of analytic changes of coordinates to normal form.