In this paper a universal formula is given for the characteristic classes dual to the cycles of multisingularities of holomorphic maps in terms of the so-called residual polynomials. The existence theorem of such a universal formula generalizes the existence theorem for the Thom polynomial to the case of multisingularities. An analogue of this formula for the case of Legendre singularities is given. The residual polynomials of singularities of low codimension are computed. In particular, applications of the formula give generalizations to the case n > 3 of the classical results of Plücker and Salmon on enumeration of singularities of tangency of a smooth hypersurface in ℂP n to projective subspaces.