  We show that most classes of shift-symmetric degenerate higher-order
scalar-tensor (DHOST) theories which satisfy certain degeneracy conditions are
not compatible with the conditions for the existence of exact black hole
solutions with a linearly time-dependent scalar field whose canonical kinetic
term takes a constant value. Combined with constraints from the propagation
speed of gravitational waves, our results imply that cubic DHOST theories are
strongly disfavoured and that pure quadratic theories are likely to be the most
viable class of DHOST theories. We find exact static and spherically symmetric
(Schwarzschild and Schwarzschild-(anti-)de Sitter) black hole solutions in all
shift-symmetric higher-derivative scalar-tensor theories which contain up to
cubic order terms of the second-order derivatives of the scalar field,
especially the full class of Horndeski and Gleyzes-Langlois-Piazza-Vernizzi
theories. After deriving the conditions for the coupling functions in the DHOST
Lagrangian that allow the exact solutions, we clarify their compatibility with
the degeneracy conditions.
