The lid-driven cavity flow is a well-known benchmark problem for the validation of new numerical methods and techniques. In experimental and numerical studies with viscoelastic fluids in such lid-driven flows, purely-elastic instabilities have been shown to appear even at very low Reynolds numbers. A finite-volume viscoelastic code, using the log-conformation formulation, is used in this work to probe the effect of viscoelasticity on the appearance of such instabilities in two-dimensional lid-driven cavities for a wide range of aspect ratios (0.125 <= Lambda = height/length <= 4.0), at different Deborah numbers under creeping-flow conditions and to understand the effects of regularization of the lid velocity. The effect of the viscoelasticity on the steady-state results and on the critical conditions for the onset of the elastic instabilities are described and compared to experimental results. (C) 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).