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.