General theoretical arguments and various analytic self-similar solutions have recently shown that magnetized and rotating astrophysical outflows may become asymptotically cylindrical, in agreement with observations of cosmical jets. A notable common feature in all such self-consistent, self-similar MHD solutions is that before final cylindrical collimation is achieved, the jet passes from a stage of oscillations in its radius, Mach number and other physical parameters. It is shown that under rather general assumptions this oscillatory behaviour of collimated outflows is not restricted to the few specific models examined so far, but instead seems to be a rather general physical property of an MHD outflow that starts non-cylindrically before it reaches collimation. It is concluded thence that astrophysical jets are topologically stable to small-amplitude, time-independent perturbations in their asymptotically cylindrical shape. Also, similarly to the familiar fluid instabilities, these oscillations may give rise to brightness enhancements along jets.