The acceleration technique known as Iterated Defect Correction (IDeC) for the numerical solution of singular initial value problems is investigated. IDeC based on the implicit Euler method performs satisfactorily and can thus be used for the efficient solution of singular boundary value problems with the shooting method. Higher order one-step methods like the box scheme or the trapezoidal rule cannot serve as a basic method because of a break-down of the asymptotic expansions of the global error caused by the singularity. The theoretical considerations are also supported by a comparison with extrapolation methods. Finally, it is shown that for similar reasons IDeC cannot be used for singular terminal value problems.