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Flow instabilities of non-Newtonian fluids severely hamper the quality of products during various chemical processes, such as fibre spinning, extrusion, and film blowing. The origin of extrusion instability has been studied over many decades. However, no consensus has been reached among the research community so far. In this paper, the possible cause of extrusion instabilities is explored using the finitely extensible nonlinear elastic conformation-dependent (FENE-CD) model with a nonmonotonic constitutive curve. Many well-documented experimental phenomena are reproduced in our simulations, and it could be concluded that the nonmonotonic constitutive curve plays an essential role in extrusion instabilities. In addition, the results imply that the die exit singularity may generate or magnify oscillations.