  We study analytically and numerically the stability of the pressure-less,
viscously spreading accretion ring. We show that the ring is unstable to small
non-axisymmetric perturbations. To perform the perturbation analysis of the
ring we use a stretching transformation of the time coordinate. We find that to
1st order, one-armed spiral structures, and to 2nd order additionally two-armed
spiral features may appear. Furthermore, we identify a dispersion relation
determining the instability of the ring. The theoretical results are confirmed
in several simulations, using two different numerical methods. These
computations prove independently the existence of a secular spiral instability
driven by viscosity, which evolves into persisting leading and trailing spiral
waves. Our results settle the question whether the spiral structures found in
earlier simulations of the spreading ring are numerical artifacts or genuine
instabilities.
