Synthetic gene drives may soon be used to suppress or eliminate populations of disease vectors, pathogens, invasive species, and agricultural pests. Recent proposals have focused on using Z-linked gene drives to control species with ZW sex determination, which include Lepidopteran pests, parasitic trematodes, and cane toads. These proposals include Z-linked ‘W-shredders’, which would suppress populations by cleaving the W chromosome and causing females to produce only sons, as well as Z-linked female-sterilizing gene drives. Here, I use eco-evolutionary simulations to evaluate the potential of some proposed Z-linked gene drives, and to produce recommendations regarding their design and use. The simulations show that W-shredders are likely to be highly effective at eradicating populations provided that resistance to W-shredding cannot evolve. However, W-shredder alleles can invade populations from very low frequencies, making it difficult to eliminate specific populations while leaving nearby populations untouched; this issue may restrict their possible uses.