  We consider minimizers of \[ F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) +
|\Omega|, \] where $F$ is a function nondecreasing in each parameter, and
$\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. This
includes, in particular, functions $F$ which depend on just some of the first
$N$ eigenvalues, such as the often studied $F=\lambda_N$. The existence of a
minimizer, which is also a bounded set of finite perimeter, was shown recently.
Here we show that the reduced boundary of the minimizers $\Omega$ is made up of
smooth graphs, and examine the difficulties in classifying the singular points.
Our approach is based on an approximation ("vanishing viscosity") argument,
which--counterintuitively--allows us to recover an Euler-Lagrange equation for
the minimizers which is not otherwise available.
