In fair division, local envy-freeness is a desirable property which has been thoroughly studied in recent years. In this paper, we study explanations which can be given to explain that no allocation of items can satisfy this criterion, in the house allocation setting where agents receive a single item. While MUSes are key concepts to extract explanations, they cannot be used as such: (i) they highly depend on the initial encoding of the problem; (ii) they are flat structures which fall short of capturing the dynamics of explanations; (iii) they typically come in large number and exhibit great diversity. In this paper we provide two SAT encodings of the problem which allows us to extract MUS when instances are unsatisfiable. We build a dynamic graph structure which allows to follow step-by-step the explanation. Finally, we propose several criteria to select MUSes, some of them being based on the MUS structure, while others rely on this original graphical explanation structure. We give theoretical bounds on these metrics, showing that they can vary significantly for some instances. Experimental results on synthetic data complement these results and illustrate the impact of the encodings and the relevance of our metrics to select among the many MUSes.