The aim of this paper is to investigate superresolution in deconvolution driven by sparsity priors. The observed signal is a convolution of an original signal with a continuous kernel.With the prior knowledge that the original signal can be considered as a sparse combination of Dirac delta peaks, we seek to estimate the positions and amplitudes of these peaks by solving a finite dimensional convex problem on a computational grid. Because, the support of the original signal may or may not be on this grid, by studying the discrete deconvolution of sparse peaks using L1-norm sparsity prior, we confirm recent observations that canonically the discrete reconstructions will result in multiple peaks at grid points adjacent to the location of the true peak. Owning to the complexity of this problem, we analyse carefully the de-convolution of single peaks on a grid and gain a strong insight about the dependence of the reconstructed magnitudes on the exact peak location. This in turn allows us to infer further information on recovering the location of the exact peaks i.e. to perform super-resolution. We analyze in detail the possible cases that can appear and based on our theoretical findings, we propose an self-driven adaptive grid approach that allows to perform superresolution in one-dimensional and multi-dimensional spaces. With the view that the current study can provide a further step in the development of more robust algorithms for the detection of single molecules in fluorescence microscopy or identification of characteristic frequencies in spectral analysis, we demonstrate how the proposed approach can recover sparse signals using simulated clusters of point sources (peaks) of low-resolution in one and two-dimensional spaces.