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 $n$:
$$
\begin{array}{lccccccc}
n & -3 & -2 & -1 & 0 & 1 & 2 & 3\\\hline
1 & & & & 1 & 1 & & \\
2 & & & 3 &10 & 3 & & \\
3 & & 3 & 84 & 338 & 84 & 3 & \\
4 & 60 & 1200 & 10020 & 42976 & 10020 & 1200 & 60
\end{array}
$$



Determinants of binary matrices
This question is too hard, because easier questions are already known to be hard.


The middle column is [A046747](http://oeis.org/A046747) in the OEIS, which is essentially equivalent to [A057982](http://oeis.org/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](https://arxiv.org/pdf/1812.09016.pdf).


As you mentioned yourself, the [Hadamard maximal determinant problem](http://www.indiana.edu/~maxdet/) is unsolved, with [order 15](https://arxiv.org/pdf/math/0401179v1.pdf) (for the {±1} version of the problem) being difficult already. The [determinant spectrum problem](http://www.indiana.edu/~maxdet/spectrum.html), mentioned by Gerhard Paseman, is even harder, and is easier than your question, since it's just asking for which values are zero.


