High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations (Romberg-type integration formulas). They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm equations with periodic kernel and right-hand side. The use of the quadrature methods is demonstrated by numerical examples.