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General relativity (GR) is a theory of gravity that has been tested on various scales with high precision. However, this theory is only valid in low-energy frameworks : when one tries to quantify the theory in order to describe the quantum nature of gravity, one encounters several issues. Therefore, it is relevant to look for extensions of GR that would be effectively valid up to a higher energy ; a simple way to do such an extension is to add a scalar field to the Einstein-Hilbert action, leading to the so-called scalar-tensor theories of gravity. These theories have a lot more freedom than GR when it comes to background solutions : it is therefore relevant to study the stability of such solutions in order to rule out the ones that exhibit unstable behaviour. To do so, we perturb the metric around a general background solution of a cubic Horndeski scalar-tensor theory and compute the effective metric in which odd parity perturbations propagate. This allows us to recover several stability criteria already obtained in the literature. We also compute this effective metric for some existing black hole solutions and show that these exhibit properties that are unexpected when looking only at the background : new naked singularities, shifting of the horizon...