Neutrinos are one of the most promising messengers for signals of new physics Beyond the Standard Model (BSM). On the theoretical side, their elusive nature, combined with their unknown mass mechanism, seems to indicate that the neutrino sector is indeed opening a window to new physics. On the experimental side, several long-standing anomalies have been reported in the past decades, providing a strong motivation to thoroughly test the standard three-neutrino oscillation paradigm. In this Snowmass21 white paper, we explore the potential of current and future neutrino experiments to explore BSM effects on neutrino flavor during the next decade.