A Lefschetz fibration $M^4 \to S^2$ is a generalization of a surface bundle which allows finitely many nodal singular fibers. A result of Arakelov and Parshin implies that holomorphic Lefschetz fibrations of genus $g \geq 2$ admit only finitely holomorphic sections. In this talk, we will show that no such finiteness result holds for smooth sections by giving examples of genus $g$ $ (g \geq 2)$ Lefschetz fibrations with infinitely many homotopically distinct sections. This is joint work in progress with Seraphina Lee.