It is known that ${\rm PSL}_{2}(\mathbb{F})$ can be realized for some $\mathbb{F}$ but definitely not all at present. According to David Zywina's note here, ${\rm PSL}_{2}(\mathbb{F}_{27})$ is the smallest non-abelian finite simple group for which it's not yet known if it occurs as a Galois group over $\mathbb{Q}$.

It is known that ${\rm PSL}_{2}(\mathbb{F})$ can be realized for some $\mathbb{F}$ but definitely not all at present. According to David Zywina's note here, ${\rm PSL}_{2}(\mathbb{F}_{27})$ is the smallest non-abelian finite simple group for which it's not yet known if it occurs as a Galois group over $\mathbb{Q}$.

EDIT: As noted by the OP, David Zywina's paper gives realizations of ${\rm PSL}_{2}(\mathbb{F}_{\ell^{3}})$ for $\ell \equiv \pm 2, \pm 3, \pm 4 \text{ or } \pm 6 \pmod{13}$. This still leave the case of ${\rm PSL}_{2}(\mathbb{F}_{125})$ unresolved. And now the smallest order finite simple group which is not yet known to occur as a Galois group over $\mathbb{Q}$ is $^{2}B_{2}(8)$, the smallest Suzuki group, which has order $29120$. (An approach to solving the inverse Galois problem for this group is discussed on page 32 of William Chen's paper here.)