  We describe a seemingly unnoticed feature of the text-book Maxwell-Lorentz
system of classical electrodynamics which challenges its formulation in terms
of an initial value problem. For point-charges, even after appropriate
renormalization, we demonstrate that most of the generic initial data evolves
to develop singularities in the electromagnetic fields along the light cones of
the initial charge positions. We provide explicit formulas for the
corresponding fields, demonstrate how this phenomenon renders the initial value
problem ill-posed, and show how such bad initial data can be ruled out by extra
conditions in addition to the Maxwell constraints. These extra conditions,
however, require knowledge of the history of the solution and, as we discuss,
effectively turn the Maxwell-Lorentz system into a system of delay equations
much like the Fokker-Schwarzschild-Tetrode equations. For extended charges such
singular light fronts persist in a smoothened form and, as we argue, yield
physically doubtful solutions. Our results also apply to some extent to
expectation values of field operators in quantum field theory.
