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Beurling-Carleson sets have appeared in a number of areas of complex analysissuch as boundary zero sets of analytic functions, inner functions withderivative in the Nevanlinna class, cyclicity in weighted Bergman spaces,Fuchsian groups of Widom-type and the corona problem in quotient Banachalgebras. After surveying these developments, we give a general definition ofBeurling-Carleson sets and discuss some of their basic properties. We show thatthe Roberts decomposition characterizes measures that do not chargeBeurling-Carleson sets.  For a positive singular measure $\mu$ on the unit circle, let $S_\mu$ denotethe singular inner function with singular measure $\mu$. In the second part ofthe paper, we use a corona-type decomposition to relate a number of propertiesof singular measures on the unit circle such as membership of $S'_\mu$ in theNevanlinna class $\mathcal N$, area conditions on level sets of $S_\mu$ andwepability. It was known that each of these properties holds for measuresconcentrated on Beurling-Carleson sets. We show that each of these propertiesimplies that $\mu$ lives on a countable union of Beurling-Carleson sets. Wealso describe partial relations involving the membership of $S'_\mu$ in theHardy 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 thatour results are optimal.  Finally, we show that measures that live on countable unions of$\alpha$-Beurling-Carleson sets are almost in bijection with nearly-maximalsolutions of $\Delta u = u^p \cdot \chi_{u > 0}$ when $p > 3$ and $\alpha =\frac{p-3}{p-1}$.

Beurling-Carleson sets, inner functions and a semi-linear equation