As far as I know, pretty much the only aspect of classical EM that's still actively controversial within the physics community is the best way to treat the radiation reaction force exerted on an accelerating classical charged point particle by its own emitted radiation. There have been various proposals for additional terms to add to the Lorentz force law to incorporation the radiation reaction, each of which poses its own difficulties. (And I believe that some people even claim that you don't need to add anything, because a radiation reaction already follows from the Lorentz force law itself.)
The treatments that I've seen have all seemed a bit arbitrary and ad hoc, because they've just been trying to resolve after the fact issues that someone (somewhat subjectively) doesn't like about the theory of EM without a radiation reaction force. These treatments work purely within a classical framework, but if you're only concerned about phenomenological accuracy, then you'd need to incorporate the quantum effects that come up in the real world at those length scales - i.e. QED. Has anyone ever systematically taken a classical limit of QED in order to derive an exact limiting form of the radiation reaction (within the classical regime, without any $\hbar$s)? Failing that, has anyone used QED to derive a phenomenological approximation, e.g. by doing a perturbative Feynman expansion out to some maximum number of loops? This seems like it might be pretty important for understanding the phenomenology of charged-particle accelerators.
(Of course, it's interesting and informative to think about the radiation reaction from a purely classical perspective as well; I'm not implying that a first-principles approach all the way from QED is the only "right" way to approach the question. Also, there may not be a single unique answer, because a given quantum theory can have multiple inequivalent classical limits, depending on the details of just what limit is taken. Note that I'm not talking about understanding the Brehmstralung radiation emitted by an accelerating charged particle, which is already well-understood within classical EM using the Lienard-Wichert potentials. I'm specifically talking about the back-reaction - if any - that that radiation exerts back on the trajectory of the charged particle in addition to the standard Lorentz force from the external EM fields.)
