Channel charting creates a low-dimensional representation of the radio environment in a self-supervised manner using manifold learning. Preserving relative spatial distances in the latent space, channel charting is well suited to support user localization. While prior work on channel charting has mainly focused on two-dimensional scenarios, real-world environments are inherently three-dimensional. In this work, we investigate two distinct three-dimensional indoor localization scenarios using simulated, but realistic ray tracing-based datasets: a factory hall with a three-dimensional spatial distribution of datapoints, and a multistory building where each floor exhibits a two-dimensional datapoint distribution. For the first scenario, we apply the concept of augmented channel charting, which combines classical localization and channel charting, to a three-dimensional setting. For the second scenario, we introduce multistory channel charting, a two-stage approach consisting of floor classification via clustering followed by the training of a dedicated expert neural network for channel charting on each individual floor, thereby enhancing the channel charting performance. In addition, we propose a novel feature engineering method designed to extract sparse features from the beamspace channel state information that are suitable for localization.