Is classical electromagnetism a dead research field? 
*

*Is classical electromagnetism a dead research field? 

*Are there any phenomena within classical electromagnetism that we have no explanation for?

 A: Applied electrodynamics isn't dead, especially when you include light: optical tweezers, orbital angular momentum multiplexing, electromagnetic simulation of high-speed PCBs etc.
One of the main remaining problem in theoretical classical electrodynamics was an adequate expression for the self force on a radiating charge without pre-acceleration and runaway solutions. Over the last decade in particular, this has been largely solved, as claimed by Rohrlich$^1$:

With the above limitations, I claim that Maxwell’s equations and the new
  equations of motion (together with the force restriction (1) or (15)) provide a complete
  basis of the new classical electrodynamics. It is internally consistent, consistent with the law of inertia (LAD is not) and free of unphysical solutions.


[1] Fritz Rohrlich, Dynamics of a charged particle, Phys. Rev. E 77, 046609 (2008)http://arxiv.org/abs/0804.4614
A: J. D. Jackson in the introductory remarks of his chapter on 'Radiation Damping, Classical Models of Charged Particles' (3rd edition), says that the problem of radiation reaction on motion of charged particles is not yet solved. He says that we know how to find motion of charged particles in given configuration of EM fields and also how to calculate EM fields due to given charge and current densities. However, when a charged particle accelerates in a field, it also radiates and we usually ignore the radiation reaction.
You can also refer to the paper 

The basic open question of classical electrodynamics. Marijan Ribarič and Luka Šušteršič. arXiv:1005.3943.

If you consider plasma physics and magnetohydrodynamics to be part of classical electrodynamics your list of open problems may grow.
A: There are still some very important open problems in the classical electromagnetism of relativistic charges, and there is indeed no satisfactory resolution of the reaction force and self-field problems for a relativistic point charge. One good resource for this is

Conservation laws and open questions of classical electrodynamics. Marijan Ribarič and Luka Šušteršič. World Scientific, 1990. Google books, Amazon.co.uk (cheap!).

They identify five main open questions in classical electromagnetism. From their preface:

(i) Within the framework of classical electrodynamics, do the electric and magnetic fields $\mathbf{E}(\mathbf{r},t)$ and  $\mathbf{B}(\mathbf{r},t)$ give the most simple description of the physical process underlying the interaction between electric charges and currents? Or, are the scalar and vector potentials  $\phi(\mathbf{r},t)$ and $\mathbf{A}(\mathbf{r},t)$ to be preferred in analogy with quantum mechanics, and are we to see in electric and magnetic fields  $\mathbf{E}(\mathbf{r},t)$ and  $\mathbf{B}(\mathbf{r},t)$ only quantities which are convenient for characterizing the Lorentz force between electric charges and currents? If so, what kind of scalar and vector potentials are the right ones?
(ii) How and where are energy, and linear and angular momentum stored by electromagnetically interacting electric charges and currents? How are they transported? Is it possible to define the flow of energy stored by the interaction of electric charges and currents so that we avoid the paradoxical results implied by the Poynting vector? E.g., must the electromagnetic energy be flowing all over the space when only steady electric charges and currents are present?
(iii) Is the intrinsic angular momentum (spin) of electromagnetic radiation a local property like energy, as suggested in analogy with photons, or a boundary effect, as suggested by the Maxwell stress-energy tensor, which takes no explicit account of it?
(iv) How in a general case can we define the total energy, and the total linear and angular momenta carried away by electromagnetic radiation without committing ourselves to any particular energy-momentum tensor of electromagnetic fields?
(v) Is there a classical equation of motion for pointlike charges such as electrons, protons, ions, etc., that takes into account the radiation reaction force and does not violate causality, nor exhibits self-acceleration causing runaway solutions; i.e., is it possible to augment continuous classical electrodynamics with the physical concept of pointlike charged particles, or, is a point charge only a common and handy computational device?

As orbifold mentions, it is tempting to dismiss most of these as problems that are naturally solved in the transition to quantum electrodynamics. However, that neglects the fact that QED involves renormalization procedures which are also pretty hard to swallow. I tend to think that exploring the classical side of the problem is one good place to look for ideas into how the quantum side might be improved.
