Blockchain technology has attracted tremendous interest from both industry and academia. It is typically used to record a public history of transactions (e.g., payment/smart contract data), but storing nonpayment/contract data in transactions has been common. The ability to store data unrelated to payment/contract such as illicit data on blockchain may be abused for malicious purposes. For example, one may use blockchain to store the data related to child pornography and copyright violations, which are publicly visible and immutable. Moreover, an immutable blockchain is not suitable for all blockchain‐based applications. So far, numerous redaction mechanisms for the mutable blockchain have been developed. In this paper, we aim at conducting a comprehensive survey that reviews and analyzes the state‐of‐the‐art redaction mechanisms. We start by giving a general presentation of blockchain and summarize the typical methods of inserting data in blockchain. Next, we discuss the challenges of designing the redaction mechanism and propose a list of evaluation criteria. Then, redaction mechanisms of the existing mutable blockchains are systemically reviewed and analyzed based on our evaluation criteria. The analyses include algorithmic overviews, performance limitations, and security vulnerabilities. Finally, the comparisons and analyses provide new insights into these mechanisms. This survey will provide developers and researchers a comprehensive view and facilitate the design of future mutable blockchains.