Manipulation of the detected backscattered Rayleigh signal inside the bandwidth of a frequency-swept optical sub-carrier propagating into an optical fiber permits an efficient localization of faults through a Fourier operator. When the bandwidth is restricted, analysis in the frequency domain can overcome the spatial resolution limitation while also inducing a high-dimensional problem. Introducing the Lasso as a signal processing technique paired with the Baseband Subcarrier Sweep (BSS) framework allows for a methodology to consistently evaluate fiber defects. In this work, a novel technique for optical fiber monitoring within the BSS framework, hereinafter called the BSS-Lasso, is proposed and 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.