In this expository chapter, we discuss the problem of the containment of a non-zero function satisfying various boundary regularity conditions in the de Branges-Rovnyak space \(\mathcal {H}(b)\) . We review what is known and highlight the connection of this problem to the Korenblum-Roberts cyclicity theory in Bergman-type spaces, the works of Khrushchev on simultaneous approximation by polynomials, and some aspects of the Beurling-Malliavin theory. New proofs of previously known results are given to emphasize these connections. As a new contribution, we fully characterize the symbols b for which the space \(\mathcal {H}(b)\) contains a non-zero function with C∞ extension to the boundary. This extends an earlier result of Dyakonov and Khavinson dealing with the case of inner b. We end the chapter by stating a few open problems.