Fully numerical schemes are presented for high precision computations of the four-dimensional integrals arising in Galerkin surface integral equation formulations. More specifically, the focal point of this paper is the singular integrals for coincident, edge adjacent and vertex adjacent planar and curvilinear triangular elements. The proposed method, dubbed as DIRECTFN, utilizes a series of variable transformations, able to cancel both weak (1/R) and strong (1/R2) singularities. In addition, appropriate interchanges in the order of the associated one-dimensional integrations result in further regularization of the overall integrals. The final integrands are analytic functions with respect to all variables involved and, hence, the integrals can be efficiently evaluated by means of simple Gaussian integration. The accuracy and convergence properties of the new schemes are demonstrated by evaluating representative weakly singular and strongly singular integrals over planar and quadratic curvilinear elements.