Understanding the effect of mechanical stress on membranes is of primary importance in biophysics. Here we use force spectroscopy AFM to quantitatively characterize the nanomechanical stability of supported lipid bilayers as a function of their chemical composition. The onset of plastic deformation reveals itself as a repetitive jump in the approaching force curve, which represents a molecular fingerprint for the bilayer mechanical stability. By systematically probing a set of chemically distinct supported lipid bilayers (SLBs), we first show that both the headgroup and tail have a decisive effect on their mechanical properties. While the mechanical stability of the probed SLBs linearly increases by 3.3 nN upon the introduction of each additional -CH2- in the chain, it exhibits a significant dependence on the phospholipid headgroup, ranging from 3 nN for DPPA to 66 nN for DPPG. Furthermore, we also quantify the reduction of the membrane mechanical stability as a function of the number of unsaturations and molecular branching in the chemical structure of the apolar tails. Finally, we demonstrate that, upon introduction of cholesterol and ergosterol, contrary to previous belief the mechanical stability of membranes not only increases linearly in the liquid phase (DLPC) but also for phospholipids present in the gel phase (DPPC). Our results are discussed in the framework of the continuum nucleation model. This work highlights the compelling effect of subtle variations in the chemical structure of phospholipid molecules on the membrane response when exposed to mechanical forces, a mechanism of common occurrence in nature.