Consider $n\times n$ matrices with entries in $\{0,1\}$. The determinants of these ranges from $0$ to the Hadamard bound $\frac{(n+1)^{\frac{n+1}2}}{2^n}$. Assume $n$ is large enough.
What does the distribution of the determinants look like? Is it normal or skewed?
What proportion of such $n\times n$ determinants is singular?
Much is known about these Hadamard matrices, this talk by William Orrick gives a good overview:
The first few moments of the determinant $\det M$ are known exactly, for large $n$ the distribution of $|\det M|$ is conjectured to be log-normal.
For the number of singular matrices, see OEIS. The large-$n$ limit of the probability that a $n\times n$ matrix is singular is conjectured to be $n^2/2^n$.
Let $A_n$ be a random matrix where each entry is either $1$ or $-1$, and $B_n$ be a random matrix where each entry is either $0$ or $1$. The distributions of the determinants of these two matrices are closely related.
Indeed, since the distribution of $A_n$ is symmetric around $0$, we can WLOG assume that the first row and column of our $A_n$ are entirely equal to $1$. Subtract the first row from each other row, and we're left with a lower right $(n-1) \times (n-1)$ matrix where each entry is either $0$ or $-2$. Pulling out a factor of $-2$ from each row (and again using the fact that the distribution is symmetric around $0$), we see that the determinant of $A_n$ has the same distribution as $2^{n-1}$ times the determinant of $B_{n-1}$.
The distribution of the determinant of $A_n$ is log-normal. This was originally stated by Girko in 1997, but Girko's proof is opaque and seems to skirt some technical details. More transparent proofs were later given by Nguyen and Vu, and by Bao, Pan, and Zhou.
