Parallelizability of 4-dimensional infrasolvmanifolds

We show that if $M$ is an orientable 4-dimensional infrasolvmanifold and either $\beta=\beta_1(M;\mathbb{Q})\geq2$ or $M$ is a $\mathbb{S}ol_0^4$- or a $\mathbb{S}ol_{m,n}^4$-manifold (with $m\not=n$) then $M$ is parallelizable. There are non-parallelizable examples with $\beta=1$ for each of the other solvable Lie geometries $\mathbb{E}^4$, $\mathbb{N}il^4$, $\mathbb{N}il^3\times\mathbb{E}^1$ and $\mathbb{S}ol^3\times\mathbb{E}^1$. We also determine which non-orientable flat 4-manifolds have a $Pin^+$- or $Pin^-$-structure, and consider briefly this question for the other cases.