In order to better understand the sources of numerical instabilities occurring in the simulation of time-dependent flows of viscoelastic fluids at high Weissenberg numbers, we have used various types of linear stability analyses. In complement to the limited number of investigations related to the well-posedness of problems involving the flows of viscoelatic fluids, linear stability studies of steady viscoelastic flows have been carried out by various authors in both theoretical and numerical fields. In contrast to most published work where the equations have only been discretized in the cross-stream direction using a formulation with the stream function, we have used the full spatial discretization with spectral elements since we are more interested in computing the eigenvalue spectra generated by the spatial and temporal discretizations than the ones inherent in the partial differential equations. Our analysis enables us to investigate the influence of the spatial discretization, the time schemes, the various operators present in the conservation and constitutive equations and boundary conditions on the linear stability of the constitutive models.