Let $f: Y\to X$ be an irregular dihedral branched cover between closed oriented topological four-manifolds. We assume the branching set $B$ is a closed oriented surface, embedded in $X$ topologically locally flatly, except for an isolated cone singularity. Denote by $K$ the link of the singularity. I give a formula for the signature of $Y$, and show that the deviation of the signature from the locally flat case can be expressed in terms of classical-type invariants of $K$. Several natural questions emerge: which knot types can arise as singularities? what is the range of the defect to the signature? can all possible signatures over a given base be realized using two-bridge knots? I'll report on some on-going work on these questions.