Snapshot Spectral Imaging (SSI) is a cutting-edge technology for enabling the efficient acquisition of the spatiospectral content of dynamic scenes using miniaturized platforms. To achieve this goal, SSI architectures associate each spatial pixel with a specific spectral band, thus introducing a critical trade-off between spatial and spectral resolution. In this paper, we propose a computational approach for the recovery of high spatial and spectral resolution content from a single or a small number of exposures. We formulate the problem in a novel framework of spectral measurement matrix completion and we develop an efficient low-rank and graph regularized method for SSI demosaicing. Furthermore, we extend state-of-the-art approaches by considering more realistic sampling paradigms that incorporate information related to the spectral profile associated with each pixel. In addition to reconstruction quality, we also investigate the impact of recovery on subsequent analysis tasks like classification using state-of-the-art convolutional neural networks. We experimentally validate the merits of the proposed recovery scheme using synthetically generated data from indoor and satellite observations and real data obtained with an IMEC visible range SSI camera.