This thesis focuses on the field of XAI (eXplainable AI), and more particularly local post-hoc interpretability paradigm, that is to say the generation of explanations for a single prediction of a trained classifier. In particular, we study a fully agnostic context, meaning that the explanation is generated without using any knowledge about the classifier (treated as a black-box) nor the data used to train it. In this thesis, we identify several issues that can arise in this context and that may be harmful for interpretability. We propose to study each of these issues and propose novel criteria and approaches to detect and characterize them. The three issues we focus on are: the risk of generating explanations that are out of distribution; the risk of generating explanations that cannot be associated to any ground-truth instance; and the risk of generating explanations that are not local enough. These risks are studied through two specific categories of interpretability approaches: counterfactual explanations and local surrogate models.