We consider a variational model for single-image super-resolution based on the assumption that the gradient of the target image is sparse. We enforce this assumption by considering both an isotropic and an anisotropic $ℓ^0$ regularisation on the image gradient combined with a quadratic data fidelity, similarly as studied in \cite{Storath14} for signal recovery problems. For the numerical realisation of the model, we propose a novel efficient ADMM splitting algorithm whose substeps solutions are computed efficiently by means of hard-thresholding and standard conjugate-gradient solvers. We test our model on highly-degraded synthetic and real-world data and quantitatively compare our results with several sparsity-promoting variational approaches as well as with state-of-the-art deep-learning techniques. Our experiments show that thanks to the $ℓ^0$ smoothing on the gradient, the super-resolved images can be used to improve the accuracy of standard segmentation algorithms for applications like QR codes and cell detection and land-cover classification problems.