When to use separation of variables in E&M? I'd really like to know if there is a fast way to recognize if separation of variables is the most appropriate way to go about solving a problem. Are there any kind of guidelines for when to use separation of variables rather than some other method?
 A: In practice, the simple answer is that it works when it does. You need to test an unkown situation at hand, by making the assumption of a separated solution $f(x,\,y,\,\cdots)=X(x)\,Y(y)\,\cdots$ and seeing what comes from that.
Pretty much all the  equations you will need for electromagnetism have been tested for separation in co-ordinates that reflect certain problem symmetries. So you should look for former works on solving these equations in a co-ordinate system that reflects your problem's symmetry. Cartesian, cylindrical and spherical co-ordinates are the obvious ones but there are also separations in other co-ordinate systems: elliptical cylindrical, bipolar cylindrical and others. Search carefully for what's been done.
A theoretical description of what's going on is that you look for the continuous symmetries of the physical system at hand. If the solution domain is $\mathscr{D}$, then you're looking for a smooth permutation $\Pi_t : \mathscr{D}\to\mathscr{D}$ that is one to one and onto and a smooth function of its one real parameter $t$. More precisely, you're looking for one-dimensional, one-parameter Lie groups of smooth permutations. Example: $\Pi_z$ is a translation through displacement $z$ of a system with translational invariance. Roughly, if such a permutation commutes with the differential operator that defines your problem, then you'll get a separation. A necessary condition for this to be so is that there must be a co-ordinate which corresponds one-to-one with the parameter of the smooth permutation group.
See:
C. P. Boyer, E. G. Kalnins, and W. Miller, Jr. :"Symmetry and Spearation of Variables for the Helmholtz and Laplace Equations"  Nagoya Math. J. Vol. 60 (1976),  pp35-80
