Abstract: The initiation of a novel neutrino physics program at the Large Hadron Collider (LHC) motivates studying the discovery potential of existing and proposed forward neutrino experiments. This requires resolving degeneracies between new predictions and uncertainties in modeling neutrino production in the forward kinematic region. Based on a broad selection of predictions for the parent hadron spectra, we parametrize the expected correlations in the spectra of neutrinos produced in their decays, and determine the highest achievable precision for their observation. This allows constraining physics both within and beyond the Standard Model, as the shape and normalization of observed neutrino interaction spectra depend on proton- and nuclear parton distribution functions and any parameters affecting the rate of high-energy neutrino interactions. In particular, combining multiple neutrino observables could lead to an experimental confirmation of the enhanced-strangeness scenario proposed to resolve the cosmic-ray muon puzzle during LHC Run 3, as well as constrain neutrino non-standard interactions. Moreover, we assess the possibility for observing neutrino trident scattering off a nucleus $N$, $\nu N\to\nu^{(\prime)} \ell^-\ell^{(\prime)+} N$, which has previously proven to be a notoriously difficult task with few reported experimental investigations and little conclusive results. We show that by measuring high-energy neutrino interactions, even a $\mathcal{O}(10~\textrm{ton})$ forward detector yields tens of di-muon trident events, while the relevant backgrounds can be reduced greatly with no effect to the signal.