Identifying directions where extreme events occur is a major challenge in multivariate extreme value analysis. In this paper, we use the concept of sparse regular variation introduced by Meyer and Wintenberger to infer the tail dependence of a random vector X. This approach relies on the Euclidean projection onto the simplex which better exhibits the sparsity structure of the tail of X than the standard methods. Our procedure based on a rigorous methodology aims at capturing clusters of extremal coordinates of X. It also includes the identification of a threshold above which the values taken by X are considered as extreme. We provide an efficient and scalable algorithm called MUSCLE and apply it on numerical experiments to highlight the relevance of our findings. Finally we illustrate our approach with wind speed data and financial return data.