ABSTRACT Planet formation via core accretion requires the production of kilometre-sized planetesimals from cosmic dust. This process must overcome barriers to simple collisional growth, for which the streaming instability (SI) is often invoked. Dust evolution is still required to create particles large enough to undergo vigorous instability. The SI has been studied primarily with single-size dust, and the role of the full evolved dust distribution is largely unexplored. We survey the polydisperse streaming instability (PSI) with physical parameters corresponding to plausible conditions in protoplanetary discs. We consider a full range of particle stopping times, generalized dust size distributions, and the effect of turbulence. We find that while the PSI grows in many cases more slowly with an interstellar power-law dust distribution than with a single size, reasonable collisional dust evolution, producing an enhancement of the largest dust sizes, produces instability behaviour similar to the monodisperse case. Considering turbulent diffusion, the trend is similar. We conclude that if fast linear growth of PSI is required for planet formation, then dust evolution producing a distribution with peak stopping times on the order of 0.1 orbits and an enhancement of the largest dust significantly above the single power-law distribution produced by a fragmentation cascade is sufficient, along with local enhancement of the dust to gas volume mass density ratio to order unity.