The time-dependent planar flow of a dilute monodisperse polymer solution is considered in the limit where inertia is negligible. The polymers are modelled as linear-locked dumbbells with non-linear hydrodynamic friction, and the flow field is chosen to contain an isolated stagnation point. For Weissenberg numbers We above a critical value, here taken to be unity, the steady flow in this geometry contains a highly viscous birefringent strand along the outgoing flow axis, of approximately uniform width, within which the polymers are fully extended. It is shown that for sufficiently high polymer concentrations this steady flow is subject to two modes of instability: for modest Weissenberg numbers (1 < We < 2) an oscillatory varicose disturbance of the strand is linearly unstable; for higher Weissenberg numbers a strand-disturbance having a sinuous component gives rise to a finite amplitude instability. The amplitude of initial disturbance required to trigger this instability is shown to be small. The parameter regimes for concentration, Weissenberg number and polymer extensibility within which each instability occurs are found to be in qualitative agreement with the opposed jets experiments of A.J. Muller, J.A. Odell and J.P. Tatham, J. Non-Newtonian Fluid Mech., 35 (1990) 231-250. The mechanism of the instability is kinematic in character: flow perturbations modify the polymer stretch and after a time delay, given by the time taken for polymers entering the flow to become fully stretched, the birefringent strand is affected. This in turn modifies the flow further, and if the polymer concentration is high enough gives an oscillation of increasing amplitude.