  (abridged) We investigate the signatures left by the cosmic neutrino
background on the clustering of matter, CDM+baryons and halos in redshift-space
using a set of more than 1000 N-body and hydrodynamical simulations with
massless and massive neutrinos. We find that the effect neutrinos induce on the
clustering of CDM+baryons in redshift-space on small scales is almost entirely
due to the change in $\sigma_8$. Neutrinos imprint a characteristic signature
in the quadrupole of the matter (CDM+baryons+neutrinos) field on small scales,
that can be used to disentangle the effect of $\sigma_8$ and $M_\nu$. We show
that the effect of neutrinos on the clustering of halos is very different, on
all scales, to the one induced by $\sigma_8$. We find that the effects of
neutrinos of the growth rate of CDM+baryons ranges from $\sim0.3\%$ to $2\%$ on
scales $k\in[0.01, 0.5]~h{\rm Mpc}^{-1}$ for neutrinos with masses $M_\nu
\leqslant 0.15$ eV. We compute the bias between the momentum of halos and the
momentum of CDM+baryon and find it to be 1 on large scales for all models with
massless and massive neutrinos considered. This point towards a velocity bias
between halos and total matter on large scales that it is important to account
for in order to extract unbiased neutrino information from velocity/momentum
surveys such as kSZ observations. We show that baryonic effects can affect the
clustering of matter and CDM+baryons in redshift-space by up to a few percent
down to $k=0.5~h{\rm Mpc}^{-1}$. We find that hydrodynamics and astrophysical
processes, as implemented in our simulations, only distort the relative effect
that neutrinos induce on the anisotropic clustering of matter, CDM+baryons and
halos in redshift-space by less than $1\%$. Thus, the effect of neutrinos in
the fully non-linear regime can be written as a transfer function with very
weak dependence on astrophysics.
