  We develop a theory of regularity for Dirac operators with uniformly locally
square-integrable operator data. This is motivated by Stahl--Totik regularity
for orthogonal polynomials and by recent developments for continuum
Schr\"odinger operators, but contains significant new phenomena. We prove that
the symmetric Martin function at $\infty$ for the complement of the essential
spectrum has the two-term asymptotic expansion $\Im \left( z - \frac{b}{2
z}\right) + o(\frac 1z)$ as $z \to i \infty$, which is seen as a thickness
statement for the essential spectrum. The constant $b$ plays the role of a
renormalized Robin constant and enters a universal inequality involving the
lower average $L^2$-norm of the operator data. However, we show that regularity
of Dirac operators is not precisely characterized by a single scalar equality
involving $b$ and is instead characterized by a family of equalities. This work
also contains a sharp Combes--Thomas estimate (root asymptotics of
eigensolutions), a study of zero counting measures, and applications to ergodic
and decaying operator data.
