We present the results of a morphological analysis of a small subset of the Spitzer Wide-Area Infrared Extragalactic survey (SWIRE) galaxy population. The analysis is based on public Advanced Camera for Surveys (ACS) data taken inside the SWIRE N1 field, which are the deepest optical high-resolution imaging available within the SWIRE fields as of today. Our reference sample includes 156 galaxies detected by both ACS and SWIRE. Among the various galaxy morphologies, we disentangle two main classes, spheroids (or bulge-dominated galaxies) and disc-dominated ones, for which we compute the number counts as a function of flux. We then limit our sample to objects with Infrared Array Camera (IRAC) fluxes brighter than 10 μJy, estimated ~90 per cent completeness limit of the SWIRE catalogues, and compare the observed counts to model predictions. We find that the observed counts of the spheroidal population agree with the expectations of a hierarchical model while a monolithic scenario predicts steeper counts. Both scenarios, however, underpredict the number of late-type galaxies. These observations show that the large majority (close to 80 per cent) of the 3.6- and 4.5-μm galaxy population, even at these moderately faint fluxes, is dominated by spiral and irregular galaxies or mergers.