Predicting the effects of mutations on protein stability is a key problem in fundamental and applied biology, still unsolved even for the relatively simple case of small, soluble, globular, monomeric, two-state-folder proteins. Many articles discuss the limitations of prediction methods and of the datasets used to train them, which result in low reliability for actual applications despite globally capturing trends. Here, we review these and other issues by analyzing one of the most detailed, carefully curated datasets of melting temperature change (ΔTm) upon mutation for proteins with high-resolution structures. After examining the composition of this dataset to discuss imbalances and biases, we inspect several of its entries assisted by an online app for data navigation and structure display and aided by a neural network that predicts ΔTm with accuracy close to that of programs available to this end. We pose that the ΔTm predictions of our network, and also likely those of other programs, account only for a baseline-like general effect of each type of amino acid substitution which then requires substantial corrections to reproduce the actual stability changes. The corrections are very different for each specific case and arise from fine structural details which are not well represented in the dataset and which, despite appearing reasonable upon visual inspection of the structures, are hard to encode and parametrize. Based on these observations, additional analyses, and a review of recent literature, we propose recommendations for developers of stability prediction methods and for efforts aimed at improving the datasets used for training. We leave our interactive interface for analysis available online at http://lucianoabriata.altervista.org/papersdata/proteinstability2021/s1626navigation.html so that users can further explore the dataset and baseline predictions, possibly serving as a tool useful in the context of structural biology and protein biotechnology research and as material for education in protein biophysics.