Electromagnetic waves generally contain three kinds of singularities called C lines, S surfaces and disclinations. These singularities are features of the transverse electric and transverse magnetic fields of the waves and all three kinds usually occur in any given wavefield. We show that in the case of nominally uniformly polarized waves, the simple line zeros predicted for interference fringes by scalar wave theory in fact have an underlying polarization structure consisting of two C lines and an S surface. In consequence, virtually all monochromatic electromagnetic waves contain polarization states ranging from right-hand circular, through linear to left-hand circular polarization. Singularities of the electric and magnetic fields are not generally coincident in space; in fact they can be separated by arbitrarily large distances. The separation of the electric and magnetic S surfaces means that there are regions where the transverse electric and transverse magnetic vectors counterrotate. C lines are probably the most significant of the singularities, since they are not only structural features of polarization, but also organize the time structure of electromagnetic waves. They play a crucial role in determining the topology of disclinations in paraxial wavefields. In pulsed electromagnetic waves all three singularities move through space. Their behaviour, including interactions between pairs of C lines, S surfaces or disclinations, which are likely to be frequent events in pulsed waves, is discussed.