This paper addresses the problem of sparse signal recovery by deep unrolling approaches. The task of sparse restoration from linearly transformed and noisy observations occurs in many applied fields, including geoscience, biochemistry, remote sensing, and biomedical data processing, and was thoroughly studied in literature. Recently, novel approaches based on ‘deep unrolling’ or ‘deep unfolding’, have been introduced. They consist in creating deep neural networks inspired from iterative algorithms initially built for penalized loss minimization. The iterations of the algorithm are recast as neural network layers. The use of deep learning frameworks ensures an efficient implementation and the possibility to learn the algorithm native hyperparameters, through the minimization of a task-oriented loss. However, for a given application, choosing an adequate iterative scheme to unroll and fine-tuning the architecture parameters remains a challenging task. In this work, our goal is to present a comprehensive comparative study of deep unrolled approaches deployed for sparse signal reconstruction. Three architectures are introduced and compared through a motivating application, arising in analytical chemistry. A reproducible Github code is provided.