  Neutrino trident scattering is a rare Standard Model process where a
charged-lepton pair is produced in neutrino-nucleus scattering. To date, only
the dimuon final-state has been observed, with around 100 total events, while
the other channels are as yet unexplored. In this work, we compute the trident
production cross section by performing a complete four-body phase space
calculation for different hadronic targets. This provides a correct estimate
both of the coherent and the diffractive contributions to these cross sections,
but also allows us to address certain inconsistencies in the literature related
to the use of the Equivalent Photon Approximation in this context. We show that
this approximation can give a reasonable estimate only for the production of
dimuon final-states in coherent scattering, being inadmissible for all other
cases considered. We provide estimates of the number and distribution of
trident events at several current and future near detector facilities subjected
to intense neutrino beams from accelerators: five liquid-argon detectors (SBND,
$\mu$BooNE, ICARUS, DUNE and $\nu$STORM), the iron detector of T2K (INGRID) and
three detectors made of composite material (MINOS, NO$\nu$A and MINER$\nu$A).
We find that for many experiments, trident measurements are an attainable goal
and a valuable addition to their near detector physics programme.
