  We introduce a complete analytical and numerical study of the modulational
instability process in a system governed by a canonical nonlinear Schr\"odinger
equation involving local, arbitrary nonlinear responses to the applied field.
In particular, our theory accounts for the recently proposed higher-order Kerr
nonlinearities, providing very simple analytical criteria for the
identification of multiple regimes of stability and instability of plane-wave
solutions in such systems. Moreover, we discuss a new parametric regime in the
higher-order Kerr response which allows for the observation of several,
alternating stability-instability windows defining a yet unexplored instability
landscape.
