We present new subsectional, singular divergence-conforming vector bases that incorporate the edge conditions for conducting wedges. The bases are of additive kind because obtained by incrementing the regular polynomial vector bases with other subsectional basis sets that model the singular behavior of the unknown vector field in the wedge neighborhood. Singular bases of this kind, complete to arbitrarily high order, are described in a unified and consistent manner for curved quadrilateral and triangular elements. The higher order basis functions are obtained as the product of lowest order functions and Silvester-Lagrange interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties are discussed and these bases are proved to be fully compatible with the standard, high-order regular vector bases used in adjacent elements. Our singular bases guarantee normal continuity along the edges of the elements allowing for the discontinuity of tangential components, adequate modelling of the divergence, and removal of spurious solutions. These singular high-order bases provide more accurate and efficient numerical solutions of surface integral problems. Several test-case problems are considered in the paper, thereby obtaining highly accurate numerical results for the current and charge density induced on 3D sharp-wedge structures. The results are compared with other solutions when available and confirm the faster convergence of these bases on wedge problems.