Singularities defining the topology for the polarization state of nonuniformly polarizedelectromagnetic beams have been a topic of both theoretical and practical interest, including improvements to remote sensing and free-space optical communications, for many years. However, atmospheric turbulence can distort the features of singularities over long propagation distances, limiting their use in many cases. One solution being considered is the reduction of spatial coherence of light, as partially coherent beams have shown increased resistance to turbulence under a broad range of situations. Work on coherence singularities of scalar fields supports this as well. However there has been relatively little work done to explore singularities of the intersection of the two phenomena of nonuniform partial coherence and nonuniformly polarized fields. Namely the singularities in the unified representation of coherence and polarization state, such as the cross spectral density matrix of nonuniformly partially polarized wavefields. In this dissertation, we use a simple model of partially polarized electromagnetic vortex beams to highlight three different ways that one can define polarization sin- gularities in scalar wavefields. One of those, projections of the cross spectral density matrix defining the beam, has not previously been discussed. We then detail the evolution of those novel partial polarization singularities and how the position and number of singularities are affected by different levels of atmospheric turbulence. We find that there are projections where the singularities persist on propagation, sug- gesting their possible use in applications. We lastly explore a potentially simpler way to express polarization and partial polarization singularities as phase singularities. It was established by Green and iv Wolf in 1953 that an electromagnetic wave can be characterized by a complex scalar potential, including its energy and momentum densities. In this paper, we show that for electromagnetic beams this scalar potential can be used to fully describe the beam’s topology. We further demonstrate that this scalar potential can be used to characterize the topology of partially polarized vector beams as well.