Multidimensional scaling (MDS) seeks an embedding of N objects in a p < N dimensional space such that inter-vector distances approximate pair-wise object dissimilarities. Despite their popularity, MDS algorithms are sensitive to outliers, yielding grossly erroneous embeddings even if few outliers contaminate the available dissimilarities. This work introduces a robust MDS approach exploiting the degree of sparsity in the outliers present. Links with compressive sampling lead to a robust MDS solver capable of coping with outliers. The novel algorithm relies on a majorization-minimization (MM) approach to minimize a regularized stress function, whereby an iterative MDS solver involving Lasso operators is obtained. The resulting scheme identifies outliers and obtains the desired embedding at a computational cost comparable to that of non-robust MDS alternatives. Numerical tests illustrate the merits of the proposed algorithm.