This paper addresses the problem of sparse signal recovery from linearly transformed and noisy measurements. We propose to adopt the recent 'deep unrolling' paradigm, which consists in creating a deep neural network inspired from an iterative algorithm 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. For a given application, choosing an adequate iterative scheme to unroll, and fine-tuning the architecture is a challenging task. In this work, we present three deep unrolled architectures dedicated to sparse signal recovery. We then perform their comprehensive comparative study, through the motivating application, arising in analytical chemistry, of peak retrieval from blurred and noisy chromatography acquisitions.