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 $n$:
$$
\begin{array}{lccccccc}
n & -3 & -2 & -1 & 0 & 1 & 2 & 3\\\hline
1 & & & & 1 & 1 & & \\
2 & & & 3 &10 & 3 & & \\
3 & & 3 & 84 & 338 & 84 & 3 & \\
4 & 60 & 1200 & 10020 & 42976 & 10020 & 1200 & 60
\end{array}
$$
Best Answer
This question is too hard, because easier questions are already known to be hard.
The middle column is A046747 in the OEIS, which is essentially equivalent to A057982, the number of singular {±1}-valued matrices. The asymptotic behavior of this number was a longstanding open problem that was just recently solved by Tikhomirov.
As you mentioned yourself, the Hadamard maximal determinant problem is unsolved, with order 15 (for the {±1} version of the problem) being difficult already. The determinant spectrum problem, mentioned by Gerhard Paseman, is even harder, and is easier than your question, since it's just asking for which values are zero.