Over at the Cafe, after reading about TWF 285, I asked more-or-less

About how many polynomials with coefficients in $\{\pm 1\}$ and of degree $d$ are irreducible?

and that's what I want to ask here.

The first no-go analysis: since if $P$ is reducible, then it is reducible mod $3$, we get that the number of reducible such is $O(\frac{d-1}{d}3^d)$; but that's clearly too large already to help much; reducing mod $2$ we can't distinguish polynomials anymore!