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.