  The present contribution contains a quite extensive theory for the stability
analysis of plane periodic waves of general Schr{\"o}dinger equations. On one
hand, we put the one-dimensional theory, or in other words the stability theory
for longitudinal perturbations, on a par with the one available for systems of
Korteweg type, including results on co-periodic spectral instability, nonlinear
co-periodic orbital stability, side-band spectral instability and linearized
large-time dynamics in relation with modulation theory, and resolutions of all
the involved assumptions in both the small-amplitude and large-period regimes.
On the other hand, we provide extensions of the spectral part of the latter to
the multi-dimensional context. Notably, we provide suitable multi-dimensional
modulation formal asymptotics, validate those at the spectral level and use
them to prove that waves are always spectrally unstable in both the
small-amplitude and the large-period regimes.
