The engineer is frequently confronted with the need to solve boundary-value problems where the first derivative, for example, of the solution is discontinuous at one or more points. Solution of such problems by ordinary application of the finite element scheme often proves unsatisfactory when the `singularity' is of the type which cannot be removed at the start of the calculation. The paper illustrates some of the consequences which arise from these ordinary solutions and then demonstrates a process of solution which makes use of a modified Rayleigh-Ritz method. The modification provides a practical and versatile mode of calculation which allows extensive exploitation of the singular functions in augmenting the piecewise polynomials of the finite element scheme. Details of tests are given which help in assessing the accuracy of the numerical results. An important engineering activity concerns the fracture mechanics study of cracked structures where prediction of safety is based upon the value of the stress intensity factor as calculated from a linear elastic analysis. The stress intensity factor is a measure of the amplitude of the dominant singularity at the ends of the crack and examples are given of its calculation using the modified Rayleigh-Ritz method.