We study the problem of characterizing Hankel-Schur multipliers and Toeplitz-Schur multipliers of Schatten-von Neumann class S_p for 0 < p < 1. We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of S_p. We also give a characterization of the Hankel-Schur multipliers of S_p whose symbols have lacunary power series. Then the results on Hankel-Schur multipliers are used to obtain a characterization of the Toeplitz-Schur multipliers of S_p. Finally, we return to Hankel-Schur multipliers and obtain new results in the case when the symbol of the Hankel matrix is a complex measure on the unit circle.