As the SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2) pandemic continues to spread, several variants of the virus, with mutations distributed all over the viral genome, are emerging. While most of the variants present mutations having little to no effects at the phenotypic level, some of these variants are spreading at a rate that suggests they may present a selective advantage. In particular, these rapidly spreading variants present specific mutations on the spike protein. These observations call for an urgent need to characterize the effects of these variants' mutations on phenotype features like contagiousness and antigenicity. With this aim, we performed molecular dynamics simulations on a selected set of possible spike variants in order to assess the stabilizing effect of particular amino acid substitutions on the molecular complex. We specifically focused on the mutations that are both characteristic of the top three most worrying variants at the moment, i.e the English, South African, and Amazonian ones, and that occur at the molecular interface between SARS-CoV-2 spike protein and its human ACE2 receptor. We characterize these variants' effect in terms of (i) residue mobility, (ii) compactness, studying the network of interactions at the interface, and (iii) variation of shape complementarity via expanding the molecular surfaces in the Zernike basis. Overall, our analyses highlighted greater stability of the three variant complexes with respect to both the wild type and two negative control systems, especially for the English and Amazonian variants. In addition, in the three variants, we investigate the effects a not-yet observed mutation in position 501 could provoke on complex stability. We found that a phenylalanine mutation behaves similarly to the English variant and may cooperate in further increasing the stability of the South African one, hinting at the need for careful surveillance for the emergence of these mutations in the population. Ultimately, we show that the proposed observables describe key features for the stability of the ACE2-spike complex and can help to monitor further possible spike variants.