Free edge stress singularities on and near the interface of dissimilar materials were investigated using a characteristic equation deduced in terms of Airy's stress function and the boundary element method. The values of the order of stress singularities determined by boundary element analyses under uniform tension agreed well with the values calculated by using the characteristic equation. The stress singularities disappear for certain combinations of wedge angles of the pair of materials. The stress distribution calculated by the boundary element method in the vicinity of the intersection of the surfaces and the interface agreed well with that obtained by using the stress function.