I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ and three with $-2$) and these are $$ \begin{bmatrix}1 & 1 & 0\\ 0 & 1 & 1\\ 1 & 0 & 1\end{bmatrix} $$ and the different permutation of this.

Hadamard's maximum determinant problem for $\{0,1\}$ asks about the largest possible determinant for matrices with entries in $\{0,1\}$ and it is known that the sequence of the maximal determinant for $n\times n$ matrices for $n=1,2,\dots$ starts with 1, 1, 2, 3, 5, 9, 32, 56, 144, 320,1458 (https://oeis.org/A003432). It is even known that the number of different matrices realizing the maximum (not by absolute value) is 1, 3, 3, 60, 3600, 529200, 75600, 195955200, (https://oeis.org/A051752).

My question is: what is the distribution of determinants of all $n\times n$ $\{0,1\}$-matrices?

Here is the data for (very) small $n$: $$ \begin{array}{lccccccc} n & -3 & -2 & -1 & 0 & 1 & 2 & 3\\\hline 1 & & & & 1 & 1 & & \\ 2 & & & 3 &10 & 3 & & \\ 3 & & 3 & 84 & 338 & 84 & 3 & \\ 4 & 60 & 1200 & 10020 & 42976 & 10020 & 1200 & 60 \end{array} $$