We report the results of SIMS isotopic analyses of carbon, nitrogen, oxygen, and silicon made on 849 small (approximately 1 micrometer) individual silicon carbide grains from the Murchison meteorite. The isotopic compositions of the major elements carbon and silicon of most grains (mainstream) are similar to those observed in larger grain studies suggesting an AGB star origin of these grains. In contrast, the trace element nitrogen shows a clear dependency on grain size. 14N/15N ratios increase with decreasing grain size, suggesting different stellar sources for grains of different size. Typically observed 14N/15N ratios in the small grains of this study are approximately 2700, clearly larger than the values expected from model calculations of AGB stars. In addition to the three dredge-up episodes characteristic for the evolution of AGB stars, extra-mixing of CNO-processed matter in low mass AGB stars appears to be a promising possibility in order to explain the high 14N/15N ratios of the small circumstellar SiC grains. A small fraction of grains shows a silicon isotopic signature not observed in larger circumstellar SiC grains from Murchison. Their stellar origin is still uncertain. The minor type A, B, Y, and X grains were found to be present at a level of a percent, which is similar to their abundance in the larger-grain SiC separates from Murchison. Oxygen isotopic compositions are normal within the experimental uncertainties of several 10%, indicating that oxygen of stellar origin is rare or even absent in the SiC grains. We conclude that most of the oxygen is a contaminant which was introduced into the SiC grains after their formation, e.g., during sample processing in the laboratory. We identified a nitride grain, most likely Si3N4 with little carbon, with highly anomalous isotopic compositions (12C/13C = 157 +/- 33, 14N/15N = 18 +/- 1, delta 29 Si = -43 +/- 56%, delta 30 Si = -271 +/- 50%). The isotopic patterns of carbon, nitrogen, and silicon resemble those of the rare SiC X grains suggesting that these two rare constituents of circumstellar matter formed in the same type of stellar source, namely, Type II supernovae.