Multi-layer multi-head self-attention mechanism is widely applied in modern neural language models. Attention redundancy has been observed among attention heads but has not been deeply studied in the literature. Using BERT-base model as an example, this paper provides a comprehensive study on attention redundancy which is helpful for model interpretation and model compression. We analyze the attention redundancy with Five-Ws and How. (What) We define and focus the study on redundancy matrices generated from pre-trained and fine-tuned BERT-base model for GLUE datasets. (How) We use both token-based and sentence-based distance functions to measure the redundancy. (Where) Clear and similar redundancy patterns (cluster structure) are observed among attention heads. (When) Redundancy patterns are similar in both pre-training and fine-tuning phases. (Who) We discover that redundancy patterns are task-agnostic. Similar redundancy patterns even exist for randomly generated token sequences. (Why'') We also evaluate influences of the pre-training dropout ratios on attention redundancy. Based on the phase-independent and task-agnostic attention redundancy patterns, we propose a simple zero-shot pruning method as a case study. Experiments on fine-tuning GLUE tasks verify its effectiveness. The comprehensive analyses on attention redundancy make model understanding and zero-shot model pruning promising.