Summary: Let \(\Omega\subset\mathbb{R}^d\) be a \(C^1\) domain or, more generally, a Lipschitz domain with small Lipschitz constant and \(A(x)\) be a \(d \times d\) uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume \(u\) is harmonic in \(\Omega\), or with greater generality \(u\) solves \(\operatorname{div}(A(x)\nabla u) = 0\) in \(\Omega\), and \(u\) vanishes on \(\Sigma = \partial\Omega\cap B\) for some ball \(B\). We study the dimension of the singular set of \(u\) in \(\Sigma\), in particular we show that there is a countable family of open balls \((B_i)_i\) such that \(u|_{B_i\cap\Omega}\) does not change sign and \(K\backslash\bigcup_i B_i\) has Minkowski dimension smaller than \(d-1-\epsilon\) for any compact \(K\subset\Sigma\). We also find upper bounds for the \((d-1)\)-dimensional Hausdorff measure of the zero set of \(u\) in balls intersecting \(\Sigma\) in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of \(\Sigma\) is bounded except for a set of Hausdorff dimension at most \(d-1-\epsilon\).