  A population of primordial black holes formed in the early Universe could
contribute to at least a fraction of the black-hole merger events detectable by
current and future gravitational-wave interferometers. With the ever-increasing
number of detections, an important open problem is how to discriminate whether
a given event is of primordial or astrophysical origin. We systematically
present a comprehensive and interconnected list of discriminators that would
allow us to rule out, or potentially claim, the primordial origin of a binary
by measuring different parameters, including redshift, masses, spins,
eccentricity, and tidal deformability. We estimate how accurately future
detectors (such as the Einstein Telescope and LISA) could measure these
quantities, and we quantify the constraining power of each discriminator for
current interferometers. We apply this strategy to the GWTC-3 catalog of
compact binary mergers. We show that current measurement uncertainties do not
allow us to draw solid conclusions on the primordial origin of individual
events, but this may become possible with next-generation ground-based
detectors.
