The streaming instability (SI) has been extensively studied in the linear and non-linear regimes as a mechanism to concentrate solids and trigger planetesimal formation in the mid-plane of protoplanetary discs. A related dust settling instability (DSI) applies to particles while settling towards the mid-plane. The DSI has previously been studied in the linear regime, with predictions that it could trigger particle clumping away from the mid-plane. This work presents a range of linear calculations and non-linear simulations, performed with FARGO3D, to assess conditions for DSI growth. We expand on previous linear analyses by including particle size distributions and performing a detailed study of the amount of background turbulence needed to stabilize the DSI. When including binned size distributions, the DSI often produces converged growth rates with fewer bins than the standard SI. With background turbulence, we find that the most favourable conditions for DSI growth are weak turbulence, characterized by alpha less than or similar to 10(-6) with intermediate-sized grains that settle from one gas scale height. These conditions could arise during a sudden decrease in disc turbulence following an accretion outburst. Ignoring background turbulence, we performed a parameter survey of local 2D DSI simulations. Particle clumping was either weak or occurred slower than particles settle. Clumping was reduced by a factor of 2 in a comparison 3D simulation. Overall, our results strongly disfavour the hypothesis that the DSI significantly promotes planetesimal formation. Non-linear simulations of the DSI with different numerical methods could support or challenge these findings.