Noble gas isotopes in presolar silicon carbide (SiC) dust grains from primitive meteorites provide, together with major element isotopic compositions, insight into the nucleosynthetic output of different types of evolved stars >4.5 Gyr ago. We report here new results from helium and neon isotopic analyses of single presolar SiC grains with sizes between 0.6 and 6.3 μm using an ultrahigh sensitivity mass spectrometer. These noble gas studies were complemented by an ion microprobe study (NanoSIMS) of Si, C, and N isotopic compositions of the same grains. About 40%, or 46 of the 110 grains analyzed, contain nucleosynthetic 22Ne and/or 4He from their parent stars above our mass spectrometer's detection limit. We discuss the possible stellar sources using isotopic ratios as constraints combined with new model predictions for low- to intermediate-mass (1.5, 2, 3, and 5 M☉) asymptotic giant branch (AGB) stars of different metallicities (1, 1/2, 1/3, and 1/6 Z☉). Most SiC grains are of the mainstream type and originated in low-mass AGB stars. We find a higher-than-expected percentage of A/B type grains, with some containing 22Ne and/or 4He. In addition, we find one noble gas-rich nova grain candidate, one supernova grain (X-type grain), and one 22Ne-rich X- or Z-type grain candidate.