  Let $G$ be a connected reductive linear algebraic group. We consider the
normal $G$-varieties with horospherical orbits. In this short note, we provide
a criterion to determine whether these varieties have at most canonical, log
canonical or terminal singularities in the case where they admit an algebraic
curve as rational quotient. This result seems to be new in the special setting
of torus actions with general orbits of codimension $1$. For the given
$G$-variety $X$, our criterion is expressed in terms of a weight function
$\omega_{X}$ that is constructed from the set of $G$-invariant valuations of
the function field $k(X)$. In the log terminal case, the generating function of
$\omega_{X}$ coincides with the stringy motivic volume of $X$. As an
application, we discuss the case of normal $k^{\star}$-surfaces.
