We present maximum growth rate charts of the Kelvin‐Helmholtz (KH) and Rayleigh‐Taylor (RT) instabilities at the dayside magnetopause (MP), considering two orientations of the interplanetary magnetic field (IMF) (due north and 30° west of north). We input parameters in the plasma depletion layer calculated from an MHD code. We study both a sharp MP transition and an MP with an attached boundary layer (“thin” and “thick” approximations, respectively). Our analysis applies to wavelengths (λ) from ∼2 × 103 km to ≤9 RE. Thin model results are as follows: For a stationary MP and due north IMF, the off‐noon, low‐latitude MP is very low shear (≤10°) and is substantially KH active. With an IMF inclined to north, extremely low shear, KH‐active regions are confined to two strips, one in each hemisphere, where short λ perturbations are generated, which propagate as surface ripples on the high‐latitude, duskside MP. For a sunward accelerating magnetopause and IMF north, a large part of the MP is unstable. With an inclined IMF, the KH+RT unstable strips are broader and growth rates are higher. Thick model results are as follows: For IMF due north and a stationary MP, the middle‐ to high‐latitude MP is stable. At middle to low latitudes, the inner edge of the boundary layer (IEBL) is active, except for a 2‐hour local time band on either side of noon. For the inclined IMF, the MP is stable for long λ, with activity for short λ confined to two strips, as before, with slightly reduced growth rates. For the IEBL, a clear dawn‐dusk asymmetry in KH activity is evident. When the MP accelerates sunward and the IMF points north, we have to consider also the λ of the perturbation. For short λ, growth rates are enhanced with respect to stationarity at both the MP and the IEBL. While there are extensive regions of negligible growth at the MP, the entire IEBL is RT+KH unstable. We give an example of a long λ perturbation where both interfaces are coupled and oscillate together. Finally, for an inclined IMF, we have at the MP unstable strips which are wider and have higher growth rates. The IEBL, by contrast, is completely destabilized, with larger growth rates than under stationary conditions.