A nonparallel linear stability analysis of a family of self-similar vortex cores which includes Long’s vortex as a particular member is performed using parabolized stability equations (PSE). The resulting streamwise variation of both the spatial growth rate and the axial wave number of the different unstable modes is compared with the results from a local spatial stability analysis which also takes into account the effects of viscosity and of the streamwise variation of the basic flow, so that the effect of the history of the disturbances on their stability is quantified. It is shown that this last effect is negligible for high Reynolds numbers, but becomes increasingly important as the Reynolds number decreases, especially for very small growth rates. The marching method used to solve the PSE is computationally much faster than the standard methods for solving the nonlinear eigenvalue problem resulting from the local stability equations. As a new result, the local spatial calculations reveal the existence of unstable counter-rotating spiral modes with negative group velocities for Type II Long’s vortices (that is, vortices with negative streamwise velocity at the axis), thus showing that these flows are subcritical in Benjamin’s sense. This kind of instability does not appear for Type I vortices, which can only sustain non-axisymmetric convective instabilities, and are therefore supercritical. Thus, the spatial stability analysis establishes a fundamental distinction between Type I and Type II Long’s vortices.