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The advent of powerful transformer-based discriminative language models and, more recently, generative GPT-family models, has led to notable advancements in natural language processing (NLP), particularly in commonsense reasoning tasks. One such task is commonsense reasoning, where performance is usually evaluated through multiple-choice question-answering benchmarks. Till date, many such benchmarks have been proposed and `leaderboards' tracking state-of-the-art performance on those benchmarks suggest that transformer-based models are approaching human-like performance. However, due to documented problems such as hallucination and bias, the research focus is shifting from merely quantifying accuracy on the task to an in-depth, context-sensitive probing of LLMs' generalization and robustness. To gain deeper insight into diagnosing these models' performance in commonsense reasoning scenarios, this thesis addresses three main studies: the generalization ability of transformer-based language models on commonsense reasoning, the trend in confidence distribution of these language models confronted with ambiguous inference tasks, and a proposed risk-centric evaluation framework for both discriminative and generative language models.