We present searches for rare or forbidden charm decays of the form $X_c^+\to h^\pm\ell^\mp\ell^{(\prime)+}$, where $X_c^+$ is a charm hadron ($D^+$, $D^+_s$, or $\Lambda_c^+$), $h^\pm$ is a pion, kaon, or proton, and $\ell^{(\prime)\pm}$ is an electron or muon. The analysis is based on $384 fb^{-1}$ of $e^+e^-$ annihilation data collected at or close to the $\Upsilon(4S)$ resonance with the BaBar detector at the SLAC National Accelerator Laboratory. No significant signal is observed for any of the 35 decay modes that are investigated. We establish 90% confidence-level upper limits on the branching fractions between $1 \times 10^{-6}$ and $44 \times 10^{-6}$ depending on the channel. In most cases, these results represent either the first limits or significant improvements on existing limits for the decay modes studied.