Solving linear regression problems based on the total least-squares (TLS) criterion has well-documented merits in various applications, where perturbations appear both in the data vector as well as in the regression matrix. Weighted and structured generalizations of the TLS approach are further motivated in several signal processing and system identification related problems. On the other hand, modern compressive sampling and variable selection algorithms account for perturbations of the data vector, but not those affecting the regression matrix. The present paper addresses also the latter by introducing a weighted and structured sparse (S-) TLS formulation to exploit a priori knowledge on both types of perturbations, and on the sparsity of the unknown vector. The resultant novel approach is further able to cope with sparse, under-determined errors-in-variables models with structured and correlated perturbations, while allowing for efficient sub-optimum solvers. Simulated tests demonstrate the approach, and especially its ability to reliably recover the support of unknown sparse vectors.