This paper studies robust event classification using imperfect real-world phasor measurement unit (PMU) data. By analyzing the real-world PMU data, we find it is challenging to directly use this dataset for event classifiers due to the low data quality observed in PMU measurements and event logs. To address these challenges, we develop a novel machine learning framework for training robust event classifiers, which consists of three main steps: data preprocessing, fine-grained event data extraction, and feature engineering. Specifically, the data preprocessing step addresses the data quality issues of PMU measurements (e.g., bad data and missing data); in the fine-grained event data extraction step, a model-free event detection method is developed to accurately localize the events from the inaccurate event timestamps in the event logs; and the feature engineering step constructs the event features based on the patterns of different event types, in order to improve the performance and the interpretability of the event classifiers. Based on the proposed framework, we develop a workflow for event classification using the real-world PMU data streaming into the system in real time. Using the proposed framework, robust event classifiers can be efficiently trained based on many off-the-shelf lightweight machine learning models. Numerical experiments using the real-world dataset from the Western Interconnection of the U.S power transmission grid show that the event classifiers trained under the proposed framework can achieve high classification accuracy while being robust against low-quality data.