The aim of this talk is to study a class of singularities of moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. By establishing the stability of the Lazarsfeld-Mukai bundle for some class of rank zero sheaves on a K3 surface, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety in the sense of Nakajima and that, via this isomorphism, the natural symplectic resolutions correspond to variations of GIT quotients of the quiver varieties. This is joint work with E. Arbarello.