Since the discovery of electricity, electric cables have become ubiquitous in human constructions, from machines to buildings. Insulators play a crucial role in ensuring the proper functioning of these cables, so it is important to monitor their possible damage, which can be caused by environmental contamination, severe temperature variations, and electrical and mechanical stress. While shunt conductance is a direct health indicator of cable insulation, measuring the cable average shunt conductance is not sufficient for the detection of localized insulator damage, since localized conductance variations are diluted over a long cable length in such measurements. The objective of this paper is to assess the feasibility of reflectometry techniques for the monitoring of insulator damage in electric cables. To this end, the estimation of localized conductance variations is investigated based on electrical measurements made at one end of a cable. To avoid estimating a large number of discretized conductance values along a long cable, the proposed method relies on sparse regression, which automatically focuses on localized conductance variations at unknown positions caused by accidental insulator damage. In order to efficiently apply sparse regression techniques, the telegrapher's equations describing electric wave propagation in cables are transformed through several steps into a simple linear regression form. Then, Lasso (Least Absolute Shrinkage and Selection Operator) regression is applied to process the voltage and current data collected at a single end of the monitored cable. Numerical simulations show the potential of this method for fast estimation of localized shunt conductance variations.