Elastic filaments and semiflexible polymers occur ubiquitously in biophysical systems and are key components of many complex fluids, yet our understanding of their conformational dynamics under flow is incomplete. Here, we report on experimental observations of actin filaments in simple shear and characterize their various dynamical regimes from tumbling to buckling and snaking. Numerical simulations accounting for elastohydrodynamics as well as Brownian fluctuations show perfect agreement with measurements. Using a reduced-order theoretical model, we elucidate the unexplained mechanism for the transition to snaking. Our results pave the way for a better understanding of biophysical processes, as well as the rheology of sheared soft materials, and provide a theoretical framework for the exploration of the dynamics of dilute and semidilute suspensions.