Recently, an account of the linear and nonlinear analysis of the viscoelastic Taylor-Couette flow between independently rotating cylinders against axisymmetric disturbances was presented (Avgousti & Beris 1993a). However, more recent linear stability analysis has shown that for the range of geometric and kinematic parameters studied and for high enough values of flow elasticity, the critical instabilities are caused by non-axisymmetric, time-dependent disturbances (Avgousti & Beris 1993b). In this work, we calculate the bifurcating families corresponding to each one of the two possible non-axisymmetric patterns emerging at the point of criticality, namely the spirals and ribbons and determine their stability. It is shown that for a narrow gap size, for upper convected Maxwell and Oldroyd-B fluids, at least one of the non-axisymmetric families bifurcates subcritically. This result, in conjunction with the theoretical analysis of Hopf bifurcation in presence of symmetries (Golubitsky et al. 1988), implies that neither of the bifurcating families is stable. Consequently, there is a finite transition corresponding to infinitesimal changes of the flow parameters in the vicinity of the Hopf bifurcation point. Although a change in the ratio of the Deborah and Reynolds numbers has not been found to qualitatively affect this behaviour, calculations with a wider gap size have shown that both bifurcating families become supercritical. There, a Ginzburg-Landau analysis shows that the ribbons are the stable pattern. This behaviour is qualitatively similar to that seen for the newtonian fluid, but for counterrotating cylinders, albeit there, spirals have been found to be stable (Golubitsky & Langford 1988).