One-dimensional (1D) instabilities, some of them of new type, were found in simple shear and simple elongation for Giesekus, Leonov, and Larson viscoelastic constitutive equations (CEs) with single relaxation mode. Whereas 1D instabilities in shear flow were found for all three CEs, only the Larson model manifested some ID instabilities (and even nonexistence of solution) in simple extensional flows.