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Bike-sharing systems (BSSs) have become increasingly popular around the globe and have attracted a wide range of research interests. In this paper, the demand forecasting problem in BSSs is studied. Spatial and temporal features are critical for demand forecasting in BSSs, but it is challenging to extract spatiotemporal dynamics. Another challenge is to capture the relations between spatiotemporal dynamics and external factors, such as weather, day-of-week, and time-of-day. To address these challenges, we propose a multiple spatiotemporal fusion network named MSTF-Net. MSTF-Net consists of multiple spatiotemporal blocks: 3D convolutional network (3D-CNN) blocks, eidetic 3D convolutional long short-term memory networks (E3D-LSTM) blocks, and fully-connected (FC) blocks. Specifically, 3D-CNN blocks highlight extracting short-term spatiotemporal dependence in each fragment (i.e., closeness, period, and trend); E3D-LSTM blocks further extract long-term spatiotemporal dependence over all fragments; FC blocks extract nonlinear correlations of external factors. Finally, the latent representations of E3D-LSTM and FC blocks are fused to obtain the final prediction. For two real-world datasets, it is shown that MSTF-Net outperforms seven state-of-the-art models.