It is well-known that fullerenes aggregate inside lipid membranes and that increasing the concentration may lead to (lethal) membrane rupture. It is not known, however, how aggregation and rupture depend on the lipid type, what physical mechanisms control this behavior and what experimental signatures detect such changes in membranes. In this paper, we attempt to answer these questions with molecular simulations, and we show that aggregation and membrane damage depend critically on the degree of saturation of the lipid acyl chains: unsaturated bonds, or "kinks", impose a subtle but crucial compartmentalization of the bilayer into core and surface regions leading to three distinct fullerene density maxima. In contrast, when the membrane has only fully saturated lipids, fullerenes prefer to be located close to the surface under the head groups until the concentration becomes too large and the fullerenes begin clustering. No clustering is observed in membranes with unsaturated lipids. The presence of "kinks" reverses the free energy balance; although the overall free energy profiles are similar, entropy is the dominant component in unsaturated bilayers whereas enthalpy controls the fully saturated ones. Fully saturated systems show two unique signatures: 1) membrane thickness behaves non-monotonously while the area per lipid increases monotonously. We propose this as a potential reason for the observations of low fullerene concentrations being effective against bacteria. 2) The fullerene-fullerene radial distribution function (RDF) shows splitting of the second peak indicating the emergence short-range order and the importance of the second-nearest neighbor interactions. Similar second peak splitting has been reported in metal glasses.