  The Peregrine breather is widely discussed as a model for rogue waves in deep
water. We present here a detailed numerical study of perturbations of the
Peregrine breather as a solution to the nonlinear Schr\"odinger (NLS)
equations. We first address the modulational instability of the constant
modulus solution to NLS. Then we study numerically localized and nonlocalized
perturbations of the Peregrine breather in the linear and fully nonlinear
setting. It is shown that the solution is unstable against all considered
perturbations.
