  A number of works have shown that important information on the equation of
state of matter at nuclear density can be extracted from the gravitational
waves emitted by merging neutron-star binaries. We present a comprehensive
analysis of the gravitational-wave signal emitted during the inspiral, merger
and post-merger of 56 neutron-star binaries. This sample of binaries, arguably
the largest studied to date with realistic equations of state, spans across six
different nuclear-physics equations of state and ten masses, allowing us to
sharpen a number of results recently obtained on the spectral properties of the
gravitational-wave signal. Overall we find that: (i) for binaries with masses
differing no more than $20\%$, the frequency at gravitational-wave amplitude's
maximum is related quasi-universally with the tidal deformability of the two
stars; (ii) the spectral properties vary during the post-merger phase, with a
transient phase lasting a few millisecond after the merger and followed by a
quasi-stationary phase; (iii) when distinguishing the spectral peaks between
these two phases, a number of ambiguities in the identification of the peaks
disappear, leaving a simple and robust picture; (iv) using properly identified
frequencies, quasi-universal relations are found between the spectral features
and the properties of the neutron stars; (v) for the most salient peaks
analytic fitting functions can be obtained in terms of the stellar tidal
deformability or compactness. Altogether, these results support the idea that
the equation of state of nuclear matter can be constrained tightly when a
signal in gravitational waves from binary neutron stars is detected.
