Segmental models are an alternative to frame-based models for sequence prediction, where hypothesized path weights are based on entire segment scores rather than a single frame at a time. Neural segmental models are segmental models that use neural network-based weight functions. Neural segmental models have achieved competitive results for speech recognition, and their end-to-end training has been explored in several studies. In this work, we review neural segmental models, which can be viewed as consisting of a neural network-based acoustic encoder and a finite-state transducer decoder. We study end-to-end segmental models with different weight functions, including ones based on frame-level neural classifiers and on segmental recurrent neural networks. We study how reducing the search space size impacts performance under different weight functions. We also compare several loss functions for end-to-end training. Finally, we explore training approaches, including multi-stage vs. end-to-end training and multitask training that combines segmental and frame-level losses.