  Minimum residual methods such as the least-squares finite element method
(FEM) or the discontinuous Petrov--Galerkin method with optimal test functions
(DPG) usually exclude singular data, e.g., non square-integrable loads. We
consider a DPG method and a least-squares FEM for the Poisson problem. For both
methods we analyze regularization approaches that allow the use of $H^{-1}$
loads, and also study the case of point loads. For all cases we prove
appropriate convergence orders. We present various numerical experiments that
confirm our theoretical results. Our approach extends to general well-posed
second-order problems.
