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It is shown that for each separable Banach space $X$ not admitting $\ell_1$as a spreading model there is a space $Y$ having $X$ as a quotient and notadmitting any $\ell_p$ for $1 \leq p < \infty$ or $c_0$ as a spreading model.  We also include the solution to a question of W.B. Johnson and H.P. Rosenthalon the existence of a separable space not admitting as a quotient any spacewith separable dual.

On spaces admitting no $\ell_p$ or $c_0$ spreading model