A recently developed microfluidic device, the optimized shape cross-slot extensional rheometer or OSCER [S. J. Haward, M. S. N. Oliveira, M. A. Alves, and G. H. McKinley, “Optimized cross-slot flowgeometry for microfluidic extensional rheometry,” Phys. Rev. Lett.109, 128301 (Year:2012) 10.1103/PhysRevLett.109.128301], is used to investigate the stability of viscoelastic polymer solutions in an idealized planar stagnation point flow. Aqueous polymer solutions, consisting of poly(ethylene oxide) and of hyaluronic acid with various molecular weights and concentrations, are formulated in order to provide fluids with a wide range of rheological properties. Semi-dilute solutions of high molecular weight polymers provide highly viscoelastic fluids with long relaxation times, which achieve a high Weissenberg number (Wi) at flow rates for which the Reynolds number (Re) remains low; hence the elasticity number El = Wi/Re is high. Lower concentration solutions of moderate molecular weight polymers provide only weakly viscoelastic fluids in which inertia remains important and El is relatively low. Flow birefringence observations are used to visualize the nature of flow instabilities in the fluids as the volumetric flow rate through the OSCER device is steadily incremented. At low Wi and Re, all of the fluids display a steady, symmetric, and uniform “birefringent strand” of highly oriented polymer molecules aligned along the outflowing symmetry axis of the test geometry, indicating the stability of theflow field under such conditions. In fluids of El > 1, we observe steady elastic flow asymmetries beyond a critical Weissenberg number, Wi_crit, that are similar in character to those already reported in standard cross-slot geometries [e.g., P. E. Arratia, C. C. Thomas, J. Diorio, and J. P. Gollub, “Elastic instabilities ofpolymer solutions in cross-channel flow,” Phys. Rev. Lett.96, 144502 (Year:2006) 10.1103/PhysRevLett.96.144502]. However, in fluids with El < 1 we observe a sequence of time-dependent inertio-elastic instabilities beyond a critical Reynolds number, Re_crit, characterized by high frequency spatiotemporal oscillations of the birefringent strand. By plotting the critical limits of stability for the various fluids in the Wi-Re operating space, we are able to construct a stability diagram delineating the distinct steady symmetric, steady asymmetric and inertio-elastic flow regimes in this idealized planar elongational flow device.