We consider multiple-mode fretting wear in a frictional contact of elastic bodies subjected to a small-amplitude oscillation, which may include in-plane and out-of-plane translation, torsion and tilting (periodic rolling ). While the detailed kinetics of wear depends on the particular loading history and wear mechanism, the final worn shape, under some additional conditions, occurs to be universal for all types and loading and wear mechanisms. This universal form is determined solely by the radius of the permanent stick region and the maximum indentation depth during the loading cycle. We provide experimental evidence for the correctness of the theoretically predicted limiting shape. The existence of the universal limiting shape can be used for designing joints which are resistant to fretting wear.