In this paper we study the existence problem for KM-arcs in small Desarguesian planes. We establish a full classification of KMq,t-arcs for q <= 32, up to projective equivalence. We also construct a KM64,4-arc; as t=4 was the only value for which the existence of a KM64,t-arc was unknown, this fully settles the existence problem for q <= 64
