  Beurling-Carleson sets have appeared in a number of areas of complex analysis
such as boundary zero sets of analytic functions, inner functions with
derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces,
Fuchsian groups of Widom-type and the corona problem in quotient Banach
algebras. After surveying these developments, we give a general definition of
Beurling-Carleson sets and discuss some of their basic properties. We show that
the Roberts decomposition characterizes measures that do not charge
Beurling-Carleson sets.
  For a positive singular measure $\mu$ on the unit circle, let $S_\mu$ denote
the singular inner function with singular measure $\mu$. In the second part of
the paper, we use a corona-type decomposition to relate a number of properties
of singular measures on the unit circle such as membership of $S'_\mu$ in the
Nevanlinna class $\mathcal N$, area conditions on level sets of $S_\mu$ and
wepability. It was known that each of these properties holds for measures
concentrated on Beurling-Carleson sets. We show that each of these properties
implies that $\mu$ lives on a countable union of Beurling-Carleson sets. We
also describe partial relations involving the membership of $S'_\mu$ in the
Hardy space $H^p$, membership of $S_\mu$ in the Besov space $B^p$ and
$(1-p)$-Beurling-Carleson sets and give a number of examples which show that
our results are optimal.
  Finally, we show that measures that live on countable unions of
$\alpha$-Beurling-Carleson sets are almost in bijection with nearly-maximal
solutions of $\Delta u = u^p \cdot \chi_{u > 0}$ when $p > 3$ and $\alpha =
\frac{p-3}{p-1}$.
