In this paper, a novel technique for optical fiber monitoring that introduces the least absolute shrinkage and selection operator (Lasso) as a signal processing technique within the baseband subcarrier sweep (BSS) framework, called the BSS-Lasso, is proposed. The methodology is tested in simulated and real-world environments, taking into account both reflective and nonreflective events. The results show that for fiber links ranging from 2 to 15 km with up to three faults, over 80% of faults are detected within a 50-m range, and indicate that the proposed methodology significantly outperforms current state-of-the-art BSS-based supervision techniques. Finally, the BSS-Lasso allows for precise, low-cost, transmitter-embedded full characterization of optical fiber links.