On non-smooth domains, elliptic regularity for solutions of boundary value problems, measured by Sobolev norms, will only hold for a restricted set of regularity indices, due to singularities of the solution at the boundary. When the domain is Lipschitz, the intervals of indices where such regularity holds are quite well known, but the behavior at the endpoints of the intervals is not obvious. In the talk, I will describe examples of domains for which one can prove positive or negative endpoint results. This is based on a paper [Costabel 2019], which in turn uses constructions from [Jerison-Kenig 1995] and [Filonov 1997].