In many branches of applied mathematics, the evaluation of highly oscillating integrals plays a significant role. When the oscillation frequency is much more than the number of quadrature points, the numerical approximation suffers greatly in accuracy. A classic way of approximating these integrals is using Filon-type quadrature [1] which requires explicit moment calculation, which is not necessarily easy. To solve this underlying moment problem rapidly, in this paper, we propose a Filon-type quadrature for oscillatory integrals that effectively handles a variety of weak and strong singularities in the integrand. The method depends on the interpolating trigonometric polynomial approximation of the smooth non-oscillatory part of the integrand. However, if the function being approximated has a discontinuous periodic extension, such an approximation is inflicted with Gibb’s oscillations. To overcome this difficulty, a periodic extension of the function is constructed, which is then approximated by the interpolating trigonometric polynomial [2], [3]. In fact, for several special singular weights, the proposed approach yields moment problems that can be solved analytically. Additionally, our method requires only discrete functional data on a uniform grid and no explicit knowledge of the derivatives is required to approximate the integral. Numerous numerical experiments, including oscillatory integrals with algebraic, logarithmic, and Cauchy singularities, have been presented to illustrate the effectiveness of the proposed quadrature.