Using linear stability analysis, the instability characteristics are examined of both planar wakes and mixing layers subjected to rigid‐body rotation with axis of rotation perpendicular to the plane of the ambient flow. In particular, the tendency of rotation to stabilize or destabilize three‐dimensional motions is addressed. In the inviscid limit the results are consistent with the criterion established by Pedley [J. Fluid Mech.35, 97 (1969)] and Bradshaw [J. Fluid Mech.36, 177 (1969)]. Cyclonic rotation and strong anticyclonic rotation tend to stabilize three‐dimensional motions, whereas weaker anticyclonic rotation (Ro≳1) acts to destabilize these motions. This latter instability is in the form of streamwise rolls, similar to previous results obtained for boundary layer and channel flows. It is found that this instability is stronger than the coexisting Kelvin–Helmholtz instability for roughly the range 1.5<Ro<8, and its effect is maximum for Ro&bartil;2. For the case of constant ambient shear, exact solutions are obtained which give further insight into the nature of the instability.