  We analyze rare charm baryon decays within the standard model and beyond. We
identify all null test observables in unpolarized $\Lambda_c \to p \ell^+
\ell^-$, $\ell=e, \mu$ decays, and study the new physics sensitivities. We find
that the longitudinal dilepton polarization fraction $F_L$ is sensitive to
electromagnetic dipole couplings $C_{7}^{(\prime)}$, and to the right-handed
4-fermion vector coupling $C_{9}^\prime$. The forward-backward asymmetry,
$A_{\rm FB}$, due to the GIM-suppression a standard model null test already,
probes the left-handed axialvector 4-fermion coupling $C_{10}$; its
CP-asymmetry, $A_{\rm FB}^{\rm CP}$, probes CP-violating phases in $C_{10}$.
Physics beyond the standard model can induce branching ratios of dineutrino
modes $\Lambda_c \to p \nu \bar \nu $ up to a few times $10^{-5}$, and one
order of magnitude smaller if lepton flavor universality is assumed, while
standard model rates are negligible. Charged lepton flavor violation allows for
striking signals into $e^\pm \mu^\mp$ final states, up to $10^{-6}$ branching
ratios model-independently, and up to order $10^{-8}$ in leptoquark models.
Related three-body baryon decays $\Xi_c \to \Sigma \ell \ell$, $\Xi_c \to
\Lambda \ell \ell$ and $\Omega_c \to \Xi \ell \ell$ offer similar opportunities
to test the standard model with $|\Delta c|=|\Delta u|=1$ transitions.
