  We have calculated cross sections for the production of lepton pairs by a
neutrino incident on a nucleus using both the equivalent photon approximation,
and deep inelastic formalism. We find that production of mixed flavour lepton
pairs can have production cross sections as high as 35 times those of the
traditional muon pair-production process. Rates are estimated for the SHiP and
DUNE intensity frontier experiments. We find that multiple trident production
modes, some of which have never been observed, represent observable signals
over the lifetime of the detectors. Our estimates indicate that the SHiP
collaboration should be able to observe on the order of 300 trident events
given $2\cdot 10^{20}$ POT, and that the DUNE collaboration can expect
approximately 250 trident events in their near detector given 3X10^{22} POT. We
also discuss possible applications of the neutrino trident data to be collected
at SHiP and DUNE for SM and BSM physics.
