The performance of recommender systems highly impacts both music streaming platform users and the artists providing music. As fairness is a fundamental value of human life, there is increasing pressure for these algorithmic decision-making processes to be fair as well. However, many factors make recommender systems prone to biases, resulting in unfair outcomes. Furthermore, several stakeholders are involved, who may all have distinct needs requiring different fairness considerations. While there is an increasing interest in research on recommender system fairness in general, the music domain has received relatively little attention. This mini review, therefore, outlines current literature on music recommender system fairness from the perspective of each relevant stakeholder and the stakeholders combined. For instance, various works address gender fairness: one line of research compares differences in recommendation quality across user gender groups, and another line focuses on the imbalanced representation of artist gender in the recommendations. In addition to gender, popularity bias is frequently addressed; yet, primarily from the user perspective and rarely addressing how it impacts the representation of artists. Overall, this narrative literature review shows that the large majority of works analyze the current situation of fairness in music recommender systems, whereas only a few works propose approaches to improve it. This is, thus, a promising direction for future research.