Argument mining (AM) is an interdisciplinary research field focused on the automatic identification and classification of argumentative components, such as claims and premises, and the relationships between them. Recent advances in large language models (LLMs) have significantly improved the performance of argument classification compared to traditional machine learning approaches. This study presents a comprehensive evaluation of several state-of-the-art LLMs, including GPT-5.2, Llama 4, and DeepSeek, on large publicly available argument classification corpora such as Args.me and UKP. The evaluation incorporates advanced prompting strategies, including Chain-of- Thought prompting, prompt rephrasing, voting, and certainty-based classification. Both quantitative performance metrics and qualitative error analysis are conducted to assess model behavior. The best-performing model in the study (GPT-5.2) achieves a classification accuracy of 78.0% (UKP) and 91.9% (Args.me). The use of prompt rephrasing, multi-prompt voting, and certainty estimation further improves classification performance and robustness. These techniques increase the accuracy and F1 metric of the models by typically a few percentage points (from 2% to 8%). However, qualitative analysis reveals systematic failure modes shared across models, including instabilities with respect to prompt formulation, difficulties in detecting implicit criticism, interpreting complex argument structures, and aligning arguments with specific claims. This work contributes the first comprehensive evaluation that combines quantitative benchmarking and qualitative error analysis on multiple argument mining datasets using advanced LLM prompting strategies.