  Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$
vanishes on a boundary surface ball $\partial D \cap B_{5R}(0)$. We consider an
effective version of its singular set (up to boundary) $\mathcal{S}(u):=\{X\in
\overline{D}: u(X) = |\nabla u(X)| = 0\} $ and give an estimate of its
$(d-2)$-dimensional Minkowski content, which only depends on the upper bound of
some modified frequency function of $u$ centered at $0$. Such results are
already known in the interior and at the boundary of convex domains, when the
standard frequency function is monotone at every point. The novelty of our work
on Dini domains is how to compensate for the lack of such monotone quantities
at boundary as well as interior points.
