As High-Performance Computing (HPC) systems strive towards the exascale goal, studies suggest that they will experience excessive failure rates. For this reason, detecting and classifying faults in HPC systems as they occur and initiating corrective actions before they can transform into failures will be essential for continued operation. In this paper, we propose a fault classification method for HPC systems based on machine learning that has been designed specifically to operate with live streamed data. We cast the problem and its solution within realistic operating constraints of online use. Our results show that almost perfect classification accuracy can be reached for different fault types with low computational overhead and minimal delay. We have based our study on a local dataset, which we make publicly available, that was acquired by injecting faults to an in-house experimental HPC system.