Suicide memes are memes used to express suicide-related thoughts or comment on suicide-related issues. Suicide memes are increasingly common on social media, yet remain poorly understood and potentially harmful. There is an urgent need to better understand their characteristics and to develop appropriate content moderation strategies that limits users' exposure to potentially harmful content. Currently, the absence of annotated datasets of suicide memes remains a key barrier to developing and evaluating automated moderation approaches. In this paper, we introduce FigSIM, the first dataset designed for fine-grained analysis of suicide memes. The dataset consists of 1049 memes, each annotated for (1) fine-grained suicide severity levels, (2) figurative phenomena (e.g., metaphors), and (3) suicide-related content (e.g., suicide method depiction). We benchmark 16 unimodal and multimodal models across three tasks: figurative language, suicide severity, and suicide-related content detection. Overall, FigSIM demonstrates that suicide memes pose unique challenges for both modeling and content moderation. Analysis revealed biases, such as underprediction of higher suicide severity levels, especially for figurative memes. The dataset (including splits used for analyses) is publicly available. Content Warning: This paper contains suicide-related content that may be triggering.