Maximum singular value of a random $\pm 1$ matrix

http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf Theorem 5.39 (page 23) gives a non-asymptotic upper bound on the largest singular value

For what it's worth, it's relatively easy to simulate these in Mathematica, via

Manipulate[
 Histogram[
  N[SingularValueList[ 2 RandomInteger[{0, 1}, {m, n}] - 1, 1][[
      1]]] & /@ Range[10^power]], {m, 1, 10, 1}, {n, 1, 10, 
  1}, {power, 1, 6, 1}]

I've been playing around with it, and this

is a histogram for 1,000,000 tries with $m=9$, $n=5$ for the five different singular values (each in a different colour). I'm intrigued by the peaks - were you expecting that? There is also a significant portion of matrices with one zero singular value, but I am unsure whether it is due to numerical artifacts.

In relation to the first question, a recent breakthrough result of Konstantin Tikhomirov answers affirmatively an old conjecture attributed to Von Neumann, namely that for $M_n$, an $n\times n$ random matrix with independent $\pm 1$ entries

$$\mathbb{P}(M_{n}\ \text{is singular})=\left(\frac{1}{2}+o(1)\right)^n.$$

Here is the pre-print and a relevant entry in Gil Kalai's blog can be found here.