There are several possible generalizations of the definition of irreducible holomorphic symplectic manifold to the singular context (primitive symplectic variety, irreducible symplectic orbifold, irreducible symplectic variety). We review these definitions and we consider them in the lowest dimensional case, i.e. the case of the surfaces. We classify the surfaces which satisfy these definitions, showing that all of them are contractions of rational curves on K3 surfaces, but that not all the possible contractions of rational curves on K3 surfaces satisfy all these definitions. Then we consider the Hilbert scheme of points of the singular surfaces considered, and we show that in some specific cases they are irreducible symplectic orbifolds. The talk is based on a joint work with M. Penegini and A. Perego.