jmc

There are $\left(\genfrac{}{}{0ex}{}{9}{2}\right)=36$ pairs of points in $S$, and each pair determines a line. However, there are three horizontal, three vertical, and two diagonal lines that pass through three points of $S$, and these lines are each determined by three different pairs of points in $S$. Thus the number of distinct lines is $36-2\cdot 8=\boxed{20}$.