The surface-wave instability of a draining viscoelastic film is studied. The Oldroyd four-constant model is used to incorporate elastic and shear-thinning effects into the previous analysis for a Newtonian fluid. A long-wave evolution equation is derived, and used in the subsequent linear-stability analysis, dynamical system approach, and nonlinear spectral computation. Nonlinear developments of the instability are discussed, including the damping effect of shear thinning, not explained by a linear theory.