Bilayer disks and ribbons composed of a mixture of short- and long-tail phospholipids have been studied by molecular dynamics with a coarse-grained model. The effects of system composition on the edge structure, composition, and line tension were analyzed. Increases in the fraction of short-tail lipids tend to decrease the line tension (i.e., stabilize the edge) but not eliminate it. The short-tail lipid is generally enriched at the curved rim forming the bilayer edge, with an excess of 3 to 4 molecules per nanometer (relative to the bulk), but complete segregation was not observed. In all mixtures, a region depleted in the short-tail component occurs just before the edge, corresponding to a bulge in the bilayer thickness. The bulge and depletion are more prominent as the bilayer composition shifts toward a majority of short-tail lipids. In one case, a net excess of long-tail lipids at the edge was demonstrated, suggesting that certain circumstances give rise to a "segregation inversion" in which the long-tail lipid behaves as an edge stabilizer.