One of unexpected consequences of the orbibundle  Miyaoka-Yau-Sakai inequality is a bound on the maximal number of rational degree-d curves on smooth complex  K3-surfaces of given degree obtained by Miyaoka in 2009.  After recalling the necessary notions,  in my talk I will present various results concerning the question whether the above bound is sharp for  rational (resp.  smooth rational) curves on K3-surfaces of high degree. Based on joint work with M. Schuett (Hannover) and A.  Degtyarev (Ankara).