In this paper, an absolute and convective instability approach to the Kelvin-Helmholtz instability of magnetospheric surface and body/waveguide modes is presented. Rather than considering normal modes individually, the development of localized wavepackets is considered. It is shown that the time asymptotic behavior of such a wavepacket is determined by the double roots of the dispersion relation and that each double root may be identified with a normal mode, for our particular system. The dominant behavior in any reference frame is given by the double root with the largest growth rate. We consider the absolute or convective nature of the instability in the rest frame of the magnetosphere and deduce that in this reference frame wavepackets may only be absolutely unstable (growing at any fixed point in space at large time) close to the nose of the magnetosphere and are convectively unstable (moving away so that at large time there is no disturbance at any fixed point) elsewhere. The e-folding lengths of fast surface mode wavepackets are found to be small but increase around the flanks (these will become nonlinear and lead to the broadening of the low-latitude boundary layer in agreement with previous studies) [Manuel and Samson, 1993]. Fast body modes will e-fold only once as they convect around the flanks, so would be expected to remain linear in this region. Slow-mode wavepackets are also studied and are found to have very large e-folding lengths, so that their growth on the hanks will be negligible. The results are compared to a numerical simulation and excellent agreement is obtained.