Summary: For compound quadrature formulas for the numerical evaluation of \(\int^ b_ a w(x) f(x)dx\), where the function \(f\) as well as the weight function \(w\) are allowed to have singularities at the endpoints \(a\) and \(b\), we show how to preserve the order of convergence, which is obtained for smooth \(w\) and \(f\), by an appropriate choice of the meshpoints. The main emphasis is on singularities of algebraic and logarithmic type.