We present new supertaggers trained on HPSG-based treebanks. These treebanks feature high-quality annotation based on a well-developed linguistic theory and include diverse and challenging test datasets, beyond the usual WSJ section 23 and Wikipedia data. HPSG supertagging has previously relied on MaxEnt-based models. We use SVM and neural CRF- and BERT-based methods and show that both SVM and neural supertaggers achieve considerably higher accuracy compared to the baseline. Our fine-tuned BERT-based tagger achieves 97.26% accuracy on 1000 sentences from WSJ23 and 93.88% on the completely out-of-domain The Cathedral and the Bazaar (cb)). We conclude that it therefore makes sense to integrate these new supertaggers into modern HPSG parsers, and we also hope that the diverse and difficult datasets we used here will gain more popularity in the field. We contribute the complete dataset reformatted for token classification.