We describe geometric properties of {W>α}, whereW is a standard real-valued Brownian sheet, in the neighborhood of the first hitP of the level set {W>α} along a straight line or smooth monotone curveL. In such a neighborhood we use a decomposition of the formW(s, t)=α−b(s)+B(t)+x(s, t), whereb(s) andB(t) are particular diffusion processes andx(s, t) is comparatively small, to show thatP is not on the boundary of any connected component of {W>α}. Rather, components of this set form clusters nearP. An integral test for thorn-shaped neighborhoods ofL with tip atP that do not meet {W>α} is given. We then analyse the position and size of clusters and individual connected components of {W>α} near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions ofB andb and an accounting of the error termx.