We report results of a search for flavor-changing neutral current (FCNC), lepton flavor, and lepton-number violating decays of the D0 (and its antiparticle) into three and four bodies. Using data from Fermilab charm hadroproduction experiment E791, we examine modes with two leptons (muons or electrons) and a rho(0), K( *0), or straight phi vector meson or a nonresonant pi(pi), Kpi, or KK pair of pseudoscalar mesons. No evidence for any of these decays is found. Therefore, we present branching-fraction upper limits at 90% confidence level for the 27 decay modes examined (18 new).