I am relatively new to integral equations for solving time-harmonic EM scattering problems. I have read a decent number of papers on the subject, and it seems that for formulations that can support 3D closed-volume dielectric scatterers that people use EFIE (Electric-Field Integral Equations), MFIE (Magnetic Field Integral Equations), CFIE (a linear combination of EFIE and MFIE to deal with some resonance problems leading to singularities in dielectrics), MPIE (Mixed-Potential Integral Equations), or PMCHW (Poggio, Miller, Chang, Harrington, and Wu) formulations. In terms of basis functions for Method of Moments (MoM) solutions, it seems that most people use RWG functions over triangle elements. I am wondering what the advantages\disadvantages of each of these methods are, and how to go about choosing what is most appropriate for my needs.
You should definitely use CFIE over EFIE or MFIE, due to the problem of internal resonances. Basically, if you are at/near a frequency where there is a resonance in one of your scattering objects, then the EFIE or MFIE matrix may become singular/ill-conditioned. Using a linear combination of them greatly reduces the chances of singularity. I don't know much about the other formulations. I know someone that implemented CFIE with success.
In terms of basis functions, RWG (or Whitney) elements on triangular meshes is definitely the way to go, since it enforces the divergence constraints (this applies to FEM, finite differences, and other settings). It is in many ways the "right" choice of basis on the conceptual level (in frameworks like discrete exterior calculus), and it can also be generalized to quad meshes or general polygonal meshes.