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The purpose this work is to address the question of existence and regularity of solutions to a class of nonlocal elliptic problems with variable-order fractional Laplace operator and whose behaviors are complicated by the presence of singular nonlinearities. First, we prove the existence of weak solutions for a large class of data, including measures in some cases. We also obtain additional regularity properties under suitable extra assumptions. Second, we show that, in the case of measures datum, existence analysis is strongly related to the fractional capacity associated to the fractional Sobolev spaces. As a consequence, we get the natural form of the adequate “fractional gradient” when dealing with the Hamilton–Jacobi fractional equation with nonlocal gradient term in the sense of Boccardo–Gallouët–Orsina decomposition Problem.