We propose two error control techniques for numerical integrations infast multiscale collocation methods for solving Fredholm integral equations of thesecond kind with weakly singular kernels. Both techniques utilize quadratures forsingular integrals using graded points. One has a polynomial order of accuracy ifthe integrand has a polynomial order of smoothness except at the singular point andthe other has exponential order of accuracy if the integrand has an infinite orderof smoothness except at the singular point. We estimate the order of convergenceand computational complexity of the corresponding approximate solutions of theequation. We prove that the second technique preserves the order of convergence andcomputational complexity of the original collocation method. Numerical experimentsare presented to illustrate the theoretical estimates.