  Let $u$ be a non-trivial harmonic function in a domain $D\subset
\mathbb{R}^d$ which vanishes on an open set of the boundary. In a recent paper,
we showed that if $D$ is a $C^1$-Dini domain, then within the open set the
singular set of $u$, defined as $\{X\in \overline{D}: u(X) = 0 = |\nabla
u(X)|\} $, has finite $(d-2)$-dimensional Hausdorff measure. In this paper, we
show that the assumption of $C^1$-Dini domains is sharp, by constructing a
large class of non-Dini (but almost Dini) domains whose \textit{singular sets}
have infinite $\mathcal{H}^{d-2}$-measures.
