  Most of the plasma microphysics which shapes the acceleration process of
particles at collisionless shock waves takes place in the cosmic-ray precursor,
through the interaction of accelerated particles with the unshocked plasma.
Detecting directly or indirectly the synchrotron radiation of accelerated
particles in this precursor would open a new window on the microphysics of
acceleration and of collisionless shock waves. We provide analytical estimates
of the spectrum and of the polarization fraction of the synchrotron precursor
for both relativistic and non-relativistic collisionless shock fronts,
accounting for the self-generation or amplification of magnetic turbulence. In
relativistic sources, the spectrum of the precursor is harder than that of the
shocked plasma, because the upstream residence time increases with particle
energy, leading to an effectively hard spectrum of accelerated particles in the
precursor volume. At high frequencies, typically in the optical to X-ray range,
the contribution of the precursor becomes sizeable, but we find that in most
cases studied, it remains dominated by the synchrotron or inverse Compton
contribution of the shocked plasma; its contribution might be detectable only
in trans-relativistic shock waves. Non-relativistic sources offer the
possibility of spectral imaging of the precursor by viewing the shock front
edge-on. We calculate this spectro-morphological contribution for various
parameters. The synchrotron contribution is also sizeable at the highest
frequencies (X-ray range). If the turbulence is tangled in the plane transverse
to the shock front, the resulting synchrotron radiation should be nearly
maximally linearly polarized; polarimetry thus arises as an interesting tool to
reveal this precursor.[Abridged]
