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As a unique classification scheme for urban forms and functions, the local climate zone (LCZ) system provides essential general information for any studies related to urban environments, especially on a large scale. Remote sensing data-based classification approaches are the key to large-scale mapping and monitoring of LCZs. The potential of deep learning-based approaches is not yet fully explored, even though advanced convolutional neural networks (CNNs) continue to push the frontiers for various computer vision tasks. One reason is that published studies are based on different datasets, usually at a regional scale, which makes it impossible to fairly and consistently compare the potential of different CNNs for real-world scenarios. This study is based on the big So2Sat LCZ42 benchmark dataset dedicated to LCZ classification. Using this dataset, we studied a range of CNNs of varying sizes. In addition, we proposed a CNN to classify LCZs from Sentinel-2 images, Sen2LCZ-Net. Using this base network, we propose fusing multi-level features using the extended Sen2LCZ-Net-MF. With this proposed simple network architecture and the highly competitive benchmark dataset, we obtain results that are better than those obtained by the state-of-the-art CNNs, while requiring less computation with fewer layers and parameters. Large-scale LCZ classification examples of completely unseen areas are presented, demonstrating the potential of our proposed Sen2LCZ-Net-MF as well as the So2Sat LCZ42 dataset. We also intensively investigated the influence of network depth and width and the effectiveness of the design choices made for Sen2LCZ-Net-MF. Our work will provide important baselines for future CNN-based algorithm developments for both LCZ classification and other urban land cover land use classification.