This paper investigates the nature of flow in the neighbourhood of separation of a laminar boundary layer, and is based on the work of Goldstein (1948 Quart. J. Mech. Appl. Math. 1, 43), Stewartson (1958 Quart. J. Mech. Appl. Math. 11, 399), Terrill (1960 Phil. Trans. A, 253, 55) and Stewartson (1962 J.Fluid Mech. 12, 117). The problem of establishing the existence or nonexistence of a singularity at separation for incompressible two-dimensional flow is investigated in the first three of these papers, and the last mentioned finds that if heat transfer across the boundary is permitted no singularity occurs at a point of vanishing skin friction unless the heat transfer is also zero at this point. The present work examines the possibility of the non-occurrence of singularities in other physical situations including reference to three-dimensional separation. Particular problems considered include that of conefield flow of an incompressible fluid over a delta wing for which the separation line is shown to be a line of singularities, and that of compressible flow over a yawed cylinder in which case the conclusion is that the separation line is a line of regular points if the heat transfer is non-zero along its length. The problem of separation for a general three-dimensional boundary layer is considered but not resolved.