Current zero-shot anomaly detection (ZSAD) methods show remarkable success in prompting large pre-trained vision-language models to detect anomalies in a target dataset without using any dataset-specific training or demonstration. However, these methods often focus on crafting/learning prompts that capture only coarse-grained semantics of abnormality, e.g., high-level semantics like "damaged", "imperfect", or "defective" objects. They therefore have limited capability in recognizing diverse abnormality details that deviate from these general abnormal patterns in various ways. To address this limitation, we propose FAPrompt, a novel framework designed to learn Fine-grained Abnormality Prompts for accurate ZSAD. To this end, a novel Compound Abnormality Prompt learning (CAP) module is introduced in FAPrompt to learn a set of complementary, decomposed abnormality prompts, where abnormality prompts are enforced to model diverse abnormal patterns derived from the same normality semantic. On the other hand, the fine-grained abnormality patterns can be different from one dataset to another. To enhance the cross-dataset generalization, another novel module, namely Data-dependent Abnormality Prior learning (DAP), is introduced in FAPrompt to learn a sample-wise abnormality prior from abnormal features of each test image to dynamically adapt the abnormality prompts to individual test images. Comprehensive experiments on 19 real-world datasets, covering both industrial defects and medical anomalies, demonstrate that FAPrompt substantially outperforms state-of-the-art methods by at least 3%-5% in both image- and pixel-level ZSAD tasks.