Aims. We have undertaken a reassessment of the distribution of the alpha-element abundance ratios Ne/O, S/O, and Ar/O with respect to metallicity in a sample of about 1000 spectra of Galactic and extragalactic H II regions and star-forming galaxies (SFGs) of the Local Universe. We also analyse and compare different ionisation correction factor (ICF) schemes for each element in order to obtain the most confident determination of total abundances of Ne, S, and Ar. Methods. We used the DEep Spectra of Ionised REgions Database (DESIRED) Extended project (DESIRED-E), comprising about 1000 spectra of H II regions and SFGs with direct determinations for the electron temperature (Te). We homogeneously determined the physical conditions and chemical abundances for all the sample objects. We compared the Ne/O, S/O, and Ar/O ratios obtained using three different ICF schemes for each element. We also compared the abundance patterns with the predictions of a chemical evolution model of the Milky Way and stellar Ne and S abundance determinations. Results. Following a careful analysis, we conclude that one of the tested ICF schemes provides a better match to the observed behaviour of Ne/O, S/O, and Ar/O ratios. We find that the distribution of Ne/O ratios in H II regions shows a large dispersion and no clear trend with O/H, indicating that the different ICF(Ne) schemes are not able to provide correct Ne/O ratios for most of these objects. This is not the case for SFGs, which show similar linear relations with slightly positive slopes for the distributions of log(Ne/O) with respect to 12+log(O/H) or 12+log(Ne/H). The origin of this abundance pattern may be the combination of a metallicity-dependent dust depletion of O and ICF effects. The log(S/O) versus 12+log(O/H) distribution is consistent with a constant value, especially for HII regions and when we consider both types of objects (SFGs + H II regions). However, the log(S/O) versus 12+log(S/H) distribution shows a rather tight linear fit with a positive slope. This relation seems to flatten at 12+log(S/H) ≲ 6.0. We find that the observed behaviour of S/O with S/H is compatible with some contribution of S produced by Type Ia supernovae (SNe Ia). Finally, the behaviour of log(Ar/O) versus 12+log(O/H) is very similar for H II regions and SFGs and seems to be independent of the ionisation degree and the type of ICF(Ar) used, no matter whether it is based on only the ([Ar III] lines or on the combination of [Ar III] and [Ar IV] lines. The linear fit to log(Ar/O) versus 12+log(O/H) indicates a slight decrease in log(Ar/O) as 12+log(O/H) increases. However, the log(Ar/O) versus 12+log(Ar/H) relation shows an inverse trend, with a small positive slope that could indicate a small contribution of Ar from SNe Ia.