In this work, a novel technique for optical fiber monitoring that introduces the 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 non-reflective events. The results show that, for fiber links ranging from 2 to 15 km with up to 3 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.