In this work, we consider spin-
fields in Schwarzschild (anti-)de Sitter [(A)dS] black hole spacetimes. As this spacetime is different from the Ricci-flat cases, it is necessary to modify the covariant derivative to the supercovariant derivative in order to maintain the gauge symmetry, as noted in our earlier works, and this is done here by including terms related to the cosmological constant. Together with the eigenmodes for spin-
fields on an
-sphere, we derive the master radial equations, which have effective potentials that in general include an explicitly imaginary part and energy dependence. We found that for the asymptotically AdS cases the explicit imaginary dependence automatically disappears because of the negative cosmological constant. We take this case as an example and obtain the quasinormal modes by using the Horowitz-Hubeny method [Phys. Rev. D 62, 024027 (2000)].