  Physics beyond the Standard Model can manifest itself as both new light
states and heavy degrees of freedom. In this paper, we assume that the former
comprise only a sterile neutrino, $N$. Therefore, the most agnostic description
of the new physics is given by an effective field theory built upon the
Standard Model fields as well as $N$. We show that Higgs phenomenology provides
a sensitive and potentially crucial tool to constrain effective gauge
interactions of sterile neutrinos, not yet probed by current experiments. In
parallel, this motivates a range of new Higgs decay channels with clean
signatures as candidates for the next LHC runs, including $h\to
\gamma+p_T^\text{miss}$ and $h\to \gamma\gamma+p_T^\text{miss}$.
