Introduction: Soft material surfaces like fluid lipid membranes may exhibit, as their ground state, microscopic structures superimposed to their typical appearance of undulating lamellae. The development of such anomalous superstructures has to be attributed to minimization of bending energy and may even drive the formation of shapes with sharp local bends or folds.Methods: Lipid vesicles are either observed after ultrafast cryo-fixation by transmission electron microscopy or, both under normal conditions or as vitrified samples, by x-ray microscopy.Results: Cryo-TEM pictures of DOPC vesicles exhibit different types of superstructures: we observe graininess of different densities and also sharp folds in heart shaped vesicles or as garland-like deformations on vesicle surfaces. Smooth and rough vesicles show up side by side in a population. On the same vesicle surface, regions of smooth and rough appearance may coexist, sometimes with a kind of phase border separating them. Rough membrane portions seem to prefer regions of lower curvature. In x-ray microscopy we find lines of increased contrast in POPC vesicles that we interpret as stable surface folds on membranes.Conclusion: The observed features may be interpreted as the effect of higher order contributions to the bending elasticity favoring saddle curvature.