Supported lipid bilayers (SLBs) containing different types of bilayer-associated molecules (membrane-bound molecules) where one part of the molecule resides inside the lipid bilayer and another part of the molecule sticks out of the bilayer (e.g., membrane proteins) are important biophysical model systems. SLBs are commonly formed via lipid vesicle adsorption on certain surfaces (e.g., SiO(2)). However, vesicles doped with different types of (bio)molecules often do not form an SLB on the surface, and the reasons for this are not clear. Using a newly developed model of a lipid bilayer, simulations were performed to clarify the influence of the bilayer-associated molecules on vesicle adsorption and rupture. It is shown that by increasing the concentration of membrane-bound molecules in the vesicles the tendency for vesicle rupture decreases markedly and for a certain concentration rupture does not happen. The reason for this is that vesicles containing significant concentrations of such molecules tend to deform less on the surface (lower vesicle strain), especially for a significantly corrugated bilayer-surface potential. After vesicle rupture, membrane-bound molecules face either the surface or the solution in the resulting bilayer patch on the surface, depending on whether the molecules point outward or inward in the original vesicle, respectively. Vesicle surface diffusion is also studied for weak and strong surface corrugation, where vesicles are found to be almost immobile in the latter case.