  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 behaviour 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 Schr\"odinger-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 non trivial scalar hair, as well as a class of non-stealth black
holes whose metric is distinct from Schwarzschild. The knowledge of the
asymptotic behaviours of the perturbations enables us to compute numerically
quasi-normal modes, as we show explicitly for the non-stealth solutions.
Finally, the asymptotic form of the modes also signals some pathologies in the
stealth and non-stealth solutions considered here.
