  Modulation spaces were originally introduced by Feichtinger in 1983. Since
the 2000s there have been thousands of contributions using them as correct
framework; they range from PDEs, pseudodifferential operators, quantum
mechanics, signal analysis. This justifies a deep study of such spaces and the
related Wiener ones. Recently, metaplectic Wigner distributions, which contain
as special examples the $\tau$-Wigner distributions, the ambiguity function and
the Short-time Fourier transform, have proved to characterize modulation
spaces, under suitable assumptions. We investigate the metaplectic action which
is hidden in their construction and guarantees equivalent (quasi-)norms for
such spaces. We add a new result on this topic and conclude with an exhaustive
vision of these characterizations. Similar results hold for the Wiener amalgam
ones.
