Argument mining (AM) is an interdisciplinary research field that integrates insights from logic, philosophy, linguistics, rhetoric, law, psychology, and computer science. It involves the automatic identification and extraction of argumentative components, such as premises and claims, and the detection of relationships between them, such as support, attack, or neutrality. Recently, the field has advanced significantly, especially with the advent of large language models (LLMs), which have enhanced the efficiency of analyzing and extracting argument semantics compared to traditional methods and other deep learning models. There are many benchmarks for testing and verifying the quality of LLM, but there is still a lack of research and results on the operation of these models in publicly available argument classification databases. This paper presents a study of a selection of LLM's, using diverse datasets such as Args.me and UKP. The models tested include versions of GPT, Llama, and DeepSeek, along with reasoning-enhanced variants incorporating the Chain-of-Thoughts algorithm. The results indicate that ChatGPT-4o outperforms the others in the argument classification benchmarks. In case of models incorporated with reasoning capabilities, the Deepseek-R1 shows its superiority. However, despite their superiority, GPT-4o and Deepseek-R1 still make errors. The most common errors are discussed for all models. To our knowledge, the presented work is the first broader analysis of the mentioned datasets using LLM and prompt algorithms. The work also shows some weaknesses of known prompt algorithms in argument analysis, while indicating directions for their improvement. The added value of the work is the in-depth analysis of the available argument datasets and the demonstration of their shortcomings.