This paper reports off-lattice Monte Carlo simulations of highly-branched comb homopolymers weakly adsorbed on a flat, featureless surface showing only covolume and dispersion interactions with the adsorbate. A minimal coarse-grained model, described by hard spheres connected by harmonic springs, was employed. The interaction energy of the adsorbed combs and linear chains is first discussed as a function of the molecular mass and of the number of beads in contact with the surface. The molecular size is then investigated as a function of backbone length and branching density at a fixed arm size. The apparent swelling exponents of the adsorbed combs are larger than those of the corresponding linear chains, and much larger than that of the free molecules. This result indicates a surface-induced stiffening of the comb backbone, further studied through the persistence length lpers. It is found that lpers increases upon adsorption over the free-molecule value, more so the larger is the branching density. Finally, the thickness of the adsorbed layer, the surface-induced molecular anisotropy and the molecular aspect ratio are investigated as a function of branching density and molecular mass.