  The aim of this paper is to study the singularities of certain 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. For sheaves that
are pure of dimension one, we show that these moduli spaces are, locally around
a singular point, isomorphic to a quiver variety and that, via this
isomorphism, the natural symplectic resolutions correspond to variations of GIT
quotients of the quiver variety.
