The authors have previously introduced a topological invariant of currents defined on a smooth connected orientable manifold $M$ of genus $n\geqslant2$ , called the rotation homotopy class. With the help of this invariant necessary and sufficient conditions were established for the topological equivalence of minimal sets of currents containing a nonclosed recurrent trajectory. However, the question of classification (that is, the question of distinguishing all the equivalence classes and constructing standard currents in each class) remains open. The present article is devoted to the solution of the above problem, and standard minimal sets of currents are constructed in such a way that their trajectories are geodesics on$M$in a metric of constant negative curvature.Bibliography: 15 titles.