We study the linear perturbations about nonrotating black holes in the context of degenerate higher-order scalar-tensor (DHOST) theories, using a systematic approach that extracts the asymptotic behavior of perturbations (at spatial infinity and near the horizon) directly from the first-order radial differential system governing these perturbations. For axial (odd-parity) modes, this provides an alternative to the traditional approach based on a second-order Schrödinger-like equation with an effective potential, which we also discuss for completeness. For polar (even-parity) modes, which contain an additional degree of freedom in DHOST theories, and are thus more complex, we use a direct treatment of the four-dimensional first-order differential system (without resorting to a second order reformulation). We illustrate our study with two specific types of black hole solutions: “stealth” Schwarzschild black holes, with a nontrivial scalar hair, as well as a class of nonstealth black holes whose metric is distinct from Schwarzschild. The knowledge of the asymptotic behaviors of the perturbations enables us to compute numerically quasinormal modes, as we show explicitly for the nonstealth solutions. Finally, the asymptotic form of the modes also signals some pathologies in the stealth and nonstealth solutions considered here.