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Monitoring the vibration responses of structures accurately and efficiently is the key point of structural health monitoring (SHM). The monitoring of structural vibration responses depends on the sensor system. Therefore, discovering critical sensor positions is a challenging but beneficial task for SHM. Unfortunately, the predominant approaches only focus on predicting the vibration responses and lack the capability to identify meaningful sensor positions, which inevitably restricts their predictive capabilities. Furthermore, the strong linear correlation between sensors results in instability when selecting sensor locations, which leads to the difficulty of sensor optimization. To bridge this gap, we propose a sparse learning method with feature selection (SLMFS) to identify meaningful and interpretable sensor positions. In this method, to facilitate interpretation and stability, we use sparsity-inducing penalties to select the important sensors at both individual and group levels. In addition, we introduce independent regularization for stable and consistent feature selection. We also present an efficient iterative optimization algorithm to address the SLMFS, which is guaranteed to converge to the global optimum. The synthetic data, simulation data, and laboratory data are used to validate the effectiveness and accuracy of SLMFS. The results show that sensor selection and response prediction mutually reinforce each other. On the one hand, the guidance from sensors further ensures that the response prediction produces a good representation. On the other hand, the good representation enables more precise selection of target response-related sensors. Therefore, our learning method could improve the efficiency and accuracy of response prediction.