We consider the numerical approximation of elastic problems for incompressible materials, in the framework of the large deformation regime. In particular, a number of Galerkin schemes are investigated, ranging from displacement-based finite elements to mixed finite elements and NURBS-based approximations. Our focus is mainly concerned with the capability of the numerical methods under consideration to appropriately detect bifurcations and/or limit points. To this aim, we propose a couple of simple problems, for which some theoretical results about the stability range are available. We then show that several schemes, efficient and reliable in the infinitesimal elasticity regime, may fail in reproducing the stability behaviour in the large deformation context. We recognise that this failure has its root in the relaxation of the incompressibility constraint, that many methods need to introduce in order to avoid volumetric locking effects. Some numerical results are presented to confirm the theoretical considerations. This work has been developed in collaboration with F. Auricchio, L. Beirao da Veiga, A. Reali, R.L. Taylor and P. Wriggers.