Commonsense knowledge relations are crucial for advanced NLU tasks. We examine the learnability of such relations as represented in ConceptNet, taking into account their specific properties, which can make relation classification difficult: a given concept pair can be linked by multiple relation types, and relations can have multi-word arguments of diverse semantic types. We explore a neural open world multi-label classification approach that focuses on the evaluation of classification accuracy for individual relations. Based on an in-depth study of the specific properties of the ConceptNet resource, we investigate the impact of different relation representations and model variations. Our analysis reveals that the complexity of argument types and relation ambiguity are the most important challenges to address. We design a customized evaluation method to address the incompleteness of the resource that can be expanded in future work.