See The smallest singular value of deformed random rectangular matrices.

If the diagonal elements of $D$ are of order unity then the smallest singular value of $DS$ is of order $\sqrt{m}-\sqrt{n}$ with high probability. The largest singular value is of order $\sqrt{m}+\sqrt{n}$.

See The smallest singular value of deformed random rectangular matrices.

If the diagonal elements of $D$ are of order unity then the smallest singular value of $DS$ is of order $\sqrt{m}-\sqrt{n}$ with high probability. The largest singular value is of order $\sqrt{m}+\sqrt{n}$.

If there is no constraint on $D$ the upper bound is the product of the largest singular value of $D$ and the largest singular value of $S$ (the latter being of order $\sqrt{m}+\sqrt{n}$).

See The smallest singular value of deformed random rectangular matrices.

If the diagonal elements of $D$ are of order unity then the smallest singular value of $DS$ is of order $\sqrt{m}-\sqrt{n-1}$ with high probability.

See The smallest singular value of deformed random rectangular matrices.

If the diagonal elements of $D$ are of order unity then the smallest singular value of $DS$ is of order $\sqrt{m}-\sqrt{n}$ with high probability. The largest singular value is of order $\sqrt{m}+\sqrt{n}$.