We are concerned with the local linear convective instability of the incompressible boundary-layer flows over rough rotating disks for non-Newtonian fluids. Using the Carreau model for a range of shear-thinning and shear-thickening fluids, we determine, for the first time, steady-flow profiles under the partial-slip model for surface roughness. The subsequent linear stability analyses of these flows (to disturbances stationary relative to the disk) indicate that isotropic and azimuthally-anisotropic (radial grooves) surface roughness leads to the stabilisation of both shear-thinning and -thickening fluids. This is evident in the behaviour of the critical Reynolds number and growth rates of both Type I (inviscid cross flow) and Type II (viscous streamline curvature) modes of instability. The underlying physical mechanisms are clarified using an integral energy equation.