The boundary of the convex hull of $S$ consists of points with $(x, y)$ or $(y, x)=(0, \pm 3)$, $( \pm 1, \pm 3)$, and $( \pm 2, \pm 2)$. For any triangle $T$ with vertices in $S$, we can increase its area by moving a vertex not on the boundary to some point on the boundary. Thus, if $T$ has area $M$, its vertices are all on the boundary of $S$. The next step is to see (either by inspection or by noting that T has area no larger than that of an equilateral triangle inscribed in a circle of radius $\sqrt{10}$, which has area less than 13) that $M=12$. There are 16 triangles with area 12 , all congruent to one of the following three: vertices $(2,2),(1,-3)$, and $(-3,1)$; vertices $(3,-1),(-3,-1)$, and $(1,3)$; or vertices $(3,-1)$, $(-3,-1)$, and $(0,3)$.