Geometric and topological properties of phase singularity lines in three-dimensional complex scalar wavefields are discussed. In particular, their role as the intersections of the zero contour surfaces of the real and imaginary parts of the field gives numerous insights into 3D vortex topology. In addition, complex scalar wavefields (i.e. solutions of the three-dimensional Helmholtz and paraxial equations) are compared to more general complex scalar fields, including those arising naturally from algebraic geometry.