In complex scalar fields, singularities of the phase (optical vortices, wavefront dislocations) are lines in space, or points in the plane, where the wave amplitude vanishes. Phase singularities are illustrated by zeros in edge diffraction and amphidromies in the heights of the tides. In complex vector waves, there are two sorts of polarization singularity. The polarization is purely circular on lines in space or points in the plane (C singularities); these singularities have index +/- 1/2. The polarization is purely linear on lines in space for general vector fields, and surfaces in space or lines in the plane for transverse fields (L singularities); these singularities have index +/- 1. Polarization singularities (C points and L lines) are illustrated in the pattern of tidal currents.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.