Evolution in its course has found a variety of solutions to the same optimisation problem. The advent of high-throughput genomic sequencing has made available extensive data from which, in principle, one can infer the underlying structure on which biological functions rely. In this paper, we present a new method aimed at the extraction of sites encoding structural and functional properties from a set of protein primary sequences, namely a multiple sequence alignment. The method, called critical variable selection, is based on the idea that subsets of relevant sites correspond to subsequences that occur with a particularly broad frequency distribution in the dataset. By applying this algorithm to in silico sequences, to the response regulator receiver and to the voltage sensor domain of ion channels, we show that this procedure recovers not only the information encoded in single site statistics and pairwise correlations but also captures dependencies going beyond pairwise correlations. The method proposed here is complementary to statistical coupling analysis, in that the most relevant sites predicted by the two methods differ markedly. We find robust and consistent results for datasets as small as few hundred sequences that reveal a hidden hierarchy of sites that are consistent with the present knowledge on biologically relevant sites and evolutionary dynamics. This suggests that critical variable selection is capable of identifying a core of sites encoding functional and structural information in a multiple sequence alignment.