"Traditional" hydrodynamic stability studies infer stability of a flow from a computation of eigenvalues of the linearized system. While this is well justified for the Navier-Stokes equations, no rigorous result along these lines is known for general systems of partial differential equations; indeed there are counterexamples for lower order perturbations of the wave equations. This lecture will discuss how recent results on "advective" equations can be applied to creeping flows of viscoelastic fluids of Maxwell or Oldroyd type. For spatially periodic flows, stability can be reduced to the study of a) the eigenvalues, and b) a system of non-autonomous ordinary differential equations that arises from a geometric optics approximation for short waves. A more complete result for the upper convected Maxwell model will also be discussed.