  We give a complete classification of links up to clasp-pass moves, which
coincides with Habiro's $C_3$-equivalence. We also classify links up to
band-pass and band-# moves, which are versions of the usual pass- and #-move,
respectively, where each pair of parallel strands belong to the same component.
This recovers and generalizes widely a number of partial results in the study
of these local moves. The proofs make use of clasper theory.
