Detecting harmful content on social media, such as Twitter, is made difficult by the fact that the seemingly simple yes/no classification conceals a significant amount of complexity.Unfortunately, while several datasets have been collected for training classifiers in hate and offensive speech, there is a scarcity of datasets labeled with a finer granularity of target classes and specific targets. In this paper, we introduce THOS, a dataset of 8.3k tweets manually labeled with fine-grained annotations about the target of the message. We demonstrate that this dataset makes it feasible to train classifiers, based on Large Language Models, to perform classification at this level of granularity.