Résumé : Background: It is known that some well established parametrizations of the nuclear energy density functional (EDF) do not always lead to converged results for nuclei. Earlier studies point towards the existence of a qualitative link between this finding and the appearance of finite-size instabilities of symmetric nuclear matter (SNM) near saturation density when computed within the random phase approximation (RPA). Purpose: We aim to establish a stability criterion based on computationally friendly RPA calculations that can be incorporated into fitting procedures of the coupling constants of the EDF. Therefore, a quantitative and systematic connection between the impossibility to converge self-consistent calculations of nuclei and the occurrence of finite-size instabilities in SNM is investigated for the scalar-isovector (S=0, T=1) instability of the standard Skyrme EDF. Results: Tuning the coupling constant C1ρΔρ of the gradient term that triggers scalar-isovector instabilities of the standard Skyrme EDF, we find that the occurrence of instabilities in finite nuclei depends strongly on the numerical scheme used to solve the self-consistent mean-field equations. Once the critical value of the coupling constant C1ρΔρ is determined in nuclei, one can extract the corresponding lowest density ρcrit at which a pole appears at zero energy in the RPA response function. Conclusions: Instabilities of finite nuclei can be artificially hidden due to the choice of inappropriate numerical schemes or overly restrictive, e.g., spherical, symmetries. Our analysis suggests a twofold stability criterion to avoid scalar-isovector instabilities. © 2013 American Physical Society.