arXiv:2610.35664v1 Announce Type: new Abstract: Solutions to elliptic boundary value problems on three-dimensional nonsmooth domains exhibit singular behavior near edges and vertices. Using layer potential theory, we investigate boundary integral equations for the Laplace equation and characterize the leading singular exponents of the surface density on conical domains. For two canonical geometries, we derive explicit formulas for the two families of edge singular exponents on a dihedral wedge, and characterize the modewise vertex singular exponents on an axisymmetric circular cone through the zeros of associated Legendre functions and their derivatives. Building on these two prototypes and the dilation-invariance property, we formulate the associated operator pencil for general conical domains and develop a nonlinear eigenvalue solver. This solver employs the Nystr\"om method with graded meshes, logarithmically corrected trapezoidal rules, and Brent's method to compute these singular exponents accurately and efficiently. Numerical examples for trihedral cones, regular m-hedral cones, and droplet-type cones verify the effectiveness of our approach and illustrate the interactions between edge and vertex singularities.