Mussel adhesive polymers owe their ability to strongly bind to a large variety of surfaces under water to their high content of 3,4-dihydroxy-L-phenylalanine (DOPA) groups and high positive charge. In this work, we use a set of statistical copolymers that contain medium-length poly(ethylene oxide) side chains that are anchored to the surface in three different ways: by means of (i) electrostatic forces, (ii) catechol groups (as in DOPA), and (iii) the combination of electrostatic forces and catechol groups. A nanotribological scanning probe method was utilized to evaluate the wear resistance of the formed layers as a function of normal load. It was found that the combined measurement of surface topography and stiffness provided an accurate assessment of the wear resistance of such thin layers. In particular, surface stiffness maps allowed us to identify the initiation of wear before a clear topographical wear scar was developed. Our data demonstrate that the molecular and abrasive wear resistance on silica surfaces depends on the anchoring mode and follows the order catechol groups combined with electrostatic forces > catechol groups alone > electrostatic forces alone. The devised methodology should be generally applicable for evaluating wear resistance or "robustness" of thin adsorbed layers on a variety of surfaces.