Sparse and structured signal expansions on dictionaries can be obtained through explicit modeling in the coefficient domain. The originality of the present article lies in the construction and the study of generalized shrinkage operators, whose goal is to identify structured significance maps and give rise to structured thresholding. These generalize Group Lasso and the previously introduced Elitist Lasso by introducing more flexibility in the coefficient domain modeling, and lead to the notion of social sparsity. The proposed operators are studied theoretically and embedded in iterative thresholding algorithms. Moreover, a link between these operators and a convex functional is established. Numerical studies on both simulated and real signals confirm the benefits of such an approach.