Hydrodynamic instabilities occur in the motion of non-Newtonian polymeric liquids at low flow rates that are entirely absent in the corresponding motions of Newtonian fluids. We develop a dimensionless criterion that characterizes the critical conditions for onset of elastic instabilities in two-dimensional, single-phase isothermal viscoelastic flows. The new dimensionless group is analogous to classical Taylor and G\"ortler numbers characterizing inertial instabilities of Newtonian fluids and quantifies both the curvature of the streamlines and the dynamics of the non-Newtonian fluid motion.