The stability against small azimuthal perturbations of confined modes of propagation of intense short-pulse radiation governed by relativistic and charge-displacement nonlinearities in underdense plasmas is examined theoretically. In the plane of the dimensionless parameters rho 0 identical to r0 omega p,0/c and eta identical to P0/Pcr, defined by the critical power (Pcr) and the initial conditions represented by the focal radius (r0) of the incident radiation, the unperturbed plasma frequency ( omega p,0), and the peak incident power (P0), zones corresponding to stable (single-channel) and unstable (strong filamentation) regimes of propagation are established. The general finding is that large regions of stable propagation exist. The results show that for values of rho 0 sufficiently close to the dimensionless radius of the zeroth eigenmode rho c,0, the self-channelling is stable for all values of eta >1, a condition of exceptional robustness. It is also found that for the region 1< eta <or approximately=10, the propagation is stable for a very wide range of values of rho 0. In addition, the location of the boundary separating the stable and unstable zones is largely independent of the curvature of the phase front of the incident wave and weakly influenced both by the magnitudes of the azimuthal perturbations and the detailed radial profile of the incident radiation. Since self-focusing generated solely by relativistic mechanisms tends strongly to unstable behaviour in the eta >>1 regime, these results demonstrate the crucial role of the ponderomotively driven charge displacement in stabilizing the propagation. Physically, the ponderomotive radial displacement of the electrons and the contrasting inertial confinement of the ions simultaneously produce the two chief characteristics of the channels. They are the refractive self-focusing of the propagating energy arising from the displaced electrons and the spatial stability of the channels produced by the immobile electrostatic spine formed by the ions.