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 {text {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 .