Explicit example of countable transitive model of $\sf ZF$ Do we know any explicit example of a countable transitive model for $\sf ZF$ or $\sf ZFC$?

No. And yes.
$\sf ZF$, and by extension $\sf ZFC$, cannot prove their own consistency. Therefore within the confines of $\sf ZF(C)$ we cannot even prove that there are models of $\sf ZF$ to begin with.
Not only that. The statement "$\sf ZF(C)$ is consistent" is absolute between the universe and transitive models, so if there is a transitive model, not only that $\sf ZF(C)$, but also models of $\sf ZF+\operatorname{Con}(ZF)$ and much stronger theories.
So even if we assume the consistency of $\sf ZF$, we cannot prove that there are transitive models.
However! Just like $\Bbb N$ cannot be proved to exist just in $\sf PA$, we might as well take stronger theories instead of just $\sf ZFC$. And once a theory $T$, extending $\sf ZF$, proves that there exists a transitive model, then we know that there is a least $\alpha$ such that $L_\alpha$ is a model of $\sf ZFC$, and that $\alpha$ is countable. As far as explicit go, this is pretty damn explicit.
