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As experimental efforts are costly and time consuming, computational characterization of enzyme capabilities is an attractive alternative. We present and evaluate several machine-learning models to predict which of 983 distinct enzymes, as defined via the Enzyme Commission, EC, numbers, are likely to interact with a given query molecule. Our data consists of enzyme-substrate interactions from the BRENDA database. Some interactions are attributed to natural selection and involve the enzyme's natural substrates. The majority of the interactions however involve non-natural substrates, thus reflecting promiscuous enzymatic activities. We frame this enzyme promiscuity prediction problem as a multi-label classification task. We maximally utilize inhibitor and unlabelled data to train prediction models that can take advantage of known hierarchical relationships between enzyme classes. We report that a hierarchical multi-label neural network, EPP-HMCNF, is the best model for solving this problem, outperforming k-nearest neighbors similarity-based and other machine learning models. We show that inhibitor information during training consistently improves predictive power, particularly for EPP-HMCNF. We also show that all promiscuity prediction models perform worse under a realistic data split when compared to a random data split, and when evaluating performance on non-natural substrates compared to natural substrates. We provide Python code for EPP-HMCNF and other models in a repository termed EPP (Enzyme Promiscuity Prediction) at https://github.com/hassounlab/EPP.