In this paper, distributed user scheduling schemes are proposed for the multi-user MIMO-Y channel, where three $N_{T}$-antenna users ($N_{T}=2N,\,3N$) are selected from three clusters to exchange information via an $N_{R}$-antenna amplify-and-forward (AF) relay ($N_{R}=3N$), and $N\geq1$ represents the number of data stream(s) of each unicast transmission within the MIMO-Y channel. The proposed schemes effectively harvest multi-user diversity (MuD) without the need of global channel state information (CSI) or centralized computations. In particular, a novel reference signal space (RSS) is proposed to enable the distributed scheduling for both cluster-wise (CS) and group-wise (GS) patterns. The minimum user-antenna (Min-UA) transmission with $N_{T}=2N$ is first considered. Next, we consider an equal number of relay and user antenna (ER-UA) transmission with ${N_{T}=3N}$, with the aim of reducing CSI overhead as compared to Min-UA. For ER-UA transmission, the achievable MuD orders of the proposed distributed scheduling schemes are analytically derived, which proves the superiority and optimality of the proposed RSS-based distributed scheduling. These results reveal some fundamental behaviors of MuD and the performance-complexity tradeoff of user scheduling schemes in the MIMO-Y channel.