$2$-designs admitting a flag-transitive automorphism group with socle $PSL(2,q)$

$2$-designs admitting a flag-transitive automorphism group $G$ with socle $PSL(2,q)$, where $q=p^{f}\geq 4$, are investigated in both the point-primitive and point-imprimitive cases. In the latter case, a complete classification is achieved, and three known examples occur, namely: the complementary designs of $PG(3,2)$ and $PG(3,4)$, and the $2$-$(36,8,4)$ design constructed by Devillers and Praeger in [14]. In the point-primitive case, apart from the Witt-Bose-Shrikhande linear spaces of even order $q$, $48$ sporadic examples are classified. Surprisingly, one of these numerical examples is the linear space with $v=496$ and $k=4$ admitting $P\Gamma L(2,2^{5})$ as a flag-transitive automorphism group, which was missing in the 1990 classification by Buekenhout et al. [7,36,12].