Supernovae (SNe) are considered to have a major role in dust enrichment of high redshift galaxies and, due to the short lifetimes of interstellar grains, in dust replenishment of local galaxies. Here we explore how SN dust yields depend on the mass, metallicity, and rotation rate of the progenitor stars, and on the properties of the explosion. To this aim, assuming uniform mixing inside the ejecta, we quantify the dust mass produced by a sample of SN models with progenitor masses $13~M_{odot} leq M leq 120~M_{odot}$, metallicity $rm -3 leq [Fe/H] leq 0$, rotation rate $rm v_{rm rot} = 0$ and $300$~km/s, that explode with a fixed energy of $1.2 times 10^{51}$~erg (FE models) or with explosion properties calibrated to reproduce the $rm ^{56}Ni$ - $M$ relation inferred from SN observations (CE models). We find that rotation favours more efficient dust production, particularly for more massive, low metallicity stars, but that metallicity and explosion properties have the largest effects on the dust mass and its composition. In FE models, SNe with $M leq 20 - 25 ~M_{odot}$ are more efficient at forming dust: between 0.1 and 1 $M_odot$ is formed in a single explosion, with a composition dominated by silicates, carbon and magnetite grains when $rm [Fe/H] = 0$, and by carbon and magnetite grains when $rm [Fe/H] < 0$. In CE models, the ejecta are massive and metal-rich and dust production is more efficient. The dust mass increases with $M$ and it is dominated by silicates, at all [Fe/H].
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