10. In the Cartesian coordinate system, given the point set $I=\{(x, y) \mid x, y$ are integers, and $0 \leqslant x, y \leqslant 5\}$. Then the number of different squares with vertices in the set $I$ is . $\qquad$
10. 105.

It is easy to know that there are only two types of squares that meet the conditions: squares whose sides lie on lines perpendicular to the coordinate axes, called "standard squares," and squares whose sides lie on lines not perpendicular to the coordinate axes, called "oblique squares."
(1) In standard squares, the number of squares with side length $k(k=1,2$, $\cdots, 5)$ is $(6-k)^{2}$.
(2) Since the oblique squares with vertices in the point set $I$ are all inscribed squares of some standard square, we only need to consider the number of inscribed squares of standard squares.

Obviously, a square with side length $k(k=1,2, \cdots, 5)$ has $k-1$ inscribed squares.
In summary, the number of squares that meet the conditions is
$$
\sum_{k=1}^{5}\left[(6-k)^{2}+(6-k)^{2}(k-1)\right]=105(\text { squares }) .
$$