  How many squares are spanned by $n$ points in the plane? Here we study the
corresponding maximum possible number $S_{\square}(n)$ of squares and determine
the exact values for all $n\le 17$. For $18\le n\le 100$ we give lower bounds
for $S_{\square}(n)$. Besides that a few preliminary structural results are
obtained. For the related problem of the maximum possible number of isosceles
right triangles we determine the exact values for $n\le 14$ and give lower
bounds for $15\le n\le 50$.
