We present searches for rare or forbidden charm decays of the form X(c)(+) -> h(+/-)l(+/-)l((l)+), where X(c)(+) is a charm hadron (D(+), D(s)(+), or A(c)(+)), h +/- is a pion, kaon, or proton, and l((l)+/-) is an electron or muon. The analysis is based on 384 fb(-1) of e(+)e(-) annihilation data collected at or close to the gamma(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 x 10(-6) and 44 x 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.