Combining the catalogue of galaxy morphologies in the COSMOS field and the sample of Hα emitters at redshifts z = 0.4 and z = 0.84 of the HiZELS survey, we selected ∼ 220 star-forming bulgeless systems (S ́ersic index n ≤ 1.5) at both epochs. We present their star formation properties and we investigate their contribution to the star for- mation rate function (SFRF) and global star formation rate density (SFRD) at z < 1. For comparison, we also analyse Hα emitters with more structurally evolved mor- phologies that we split into two classes according to their S ́ersic index n: intermediate (1.5 < n ≤ 3) and bulge-dominated (n > 3). At both redshifts the SFRF is dominated by the contribution of bulgeless galaxies and we show that they account for more than 60% of the cosmic SFRD at z < 1. The decrease of the SFRD with redshift is common to the three morphological types but it is stronger for bulge-dominated systems. Star- forming bulgeless systems are mostly located in regions of low to intermediate galaxy densities (Σ ∼ 1 − 4 Mpc−2) typical of field-like and filament-like environments and their specific star formation rates (sSFRs) do not appear to vary strongly with local galaxy density. Only few bulgeless galaxies in our sample have high (sSFR > 10−9 yr−1) and these are mainly low-mass systems. Above M∗ ∼ 1010 M⊙ bulgeless are evolving at a “normal” rate (10−9 yr−1 < sSFR <10−10 yr−1) and in the absence of an external trigger (i.e. mergers/strong interactions) they might not be able to develop a central classical bulge.