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The contextual word embedding model, BERT, has proved its ability on downstream tasks with limited quantities of annotated data. BERT and its variants help to reduce the burden of complex annotation work in many interdisciplinary research areas, for example, legal argument mining in digital humanities. Argument mining aims to develop text analysis tools that can automatically retrieve arguments and identify relationships between argumentation clauses. Since argumentation is one of the key aspects of case law, argument mining tools for legal texts are applicable to both academic and non-academic legal research. Domain-specific BERT variants (pre-trained with corpora from a particular background) have also achieved strong performance in many tasks. To our knowledge, previous machine learning studies of argument mining on judicial case law still heavily rely on statistical models. In this paper, we provide a broad study of both classic and contextual embedding models and their performance on practical case law from the European Court of Human Rights (ECHR). During our study, we also explore a number of neural networks when being combined with different embeddings. Our experiments provide a comprehensive overview of a variety of approaches to the legal argument mining task. We conclude that domain pre-trained transformer models have great potential in this area, although traditional embeddings can also achieve strong performance when combined with additional neural network layers.