We numerically investigate the stability problem of the injection molding process. It was indicated by Bulters and Schepens [Bulters, M. and A. Schepens, The origin of the surface defect `slip-stick' on injection mouldedProducts, Proceedings of the 16th Annual Meeting of the Polymer Processing Society, Shanghai (China), pp.144--145] that surface defects of injection molded products may be attributed to a flow instability near the free surface during thefilling stage of the mold. We examine the stability of this flow using the eXtended Pom-Pom (XPP) constitutiveequations. The model allows for controlling the degree of strain hardening of the fluids without affecting the shear behaviorconsiderably. To study the linear stability characteristics of the injection molding process we use a transient finite elementalgorithm that is able to efficiently handle time dependent viscoelastic flow problems and includes a free surfacedescription to take perturbations of the computational domain into account. It is shown that the fountain flow, which is a modelflow for the injection molding process, is subject to a viscoelastic instability. If the various rheologies are compared, we observethat the onset of unstable flow can be delayed by increasing the degree of strain hardening of the fluid(by increasing the number of arms in the Pom-Pom model). The most unstable disturbance which is obtained afterexponential growth is a swirling flow near the fountain flow surface which is consistent with the experimental findings.