The excellent tracking and particle-identification capabilities of the ALICE detector allow us to fully reconstruct hadronic decays of open-charm hadrons at central rapidity and to study leptons from charm- and beauty-hadron decays at central and forward rapidities ($y$). In this contribution, we present the latest results on the production of D~mesons reconstructed via hadronic decay channels ($|y|<$0.5), open heavy flavour hadron decay electrons at mid-rapidity ($|y|<$0.8), and open heavy flavour hadron decay muons at forward rapidity (2.5$<y<$4) in pp collisions at various collision energies. Precise measurements of the transverse momentum- and $y$-differential cross sections, which provide stringent constraints for pQCD calculations, are presented, along with comparison with model expectations. Also, the dependence of open heavy flavour production on the event-multiplicity is discussed. The dependence of the D$^0$ production on the event multiplicity and spherocity in pp collisions at $\sqrt{s}$ = 7~TeV is also reported.