  Combining the catalogue of galaxy morphologies in the COSMOS field and the
sample of H$\alpha$ emitters at redshifts $z=0.4$ and $z=0.84$ of the HiZELS
survey, we selected $\sim$ 220 star-forming bulgeless systems (S\'ersic index
$n \leq 1.5$) at both epochs. We present their star formation properties and we
investigate their contribution to the star formation rate function (SFRF) and
global star formation rate density (SFRD) at $z < 1$. For comparison, we also
analyse H$\alpha$ emitters with more structurally evolved morphologies that we
split into two classes according to their S\'ersic index $n$: intermediate ($
1.5 < n \leq 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 ($\Sigma \sim 1 - 4$
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 $M_* \sim
10^{10}$ M$_{\odot}$ 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.
