To improve data availability and durability, cloud users would like to store multiple copies of their original files at servers. The multi-copy auditing technique is proposed to provide users with the assurance that multiple copies are actually stored in the cloud. However, most multi-replica solutions rely on Public Key Infrastructure (PKI), which entails heavy overhead of certificate computation and management. In this paper, we propose an efficient multi-copy dynamic integrity auditing scheme by employing certificateless signatures (named MDSS), which gets rid of expensive certificate management overhead and avoids the key escrow problem in identity-based signatures. Specifically, we improve the classic Merkle Hash Tree (MHT) to achieve batch updates for multi-copy storage, which allows the communication overhead incurred for dynamics to be independent of the replica number. To meet the flexible storage requirement, we propose a variable replica number storage strategy, allowing users to determine the replica number for each block. Based on the fact that auditors may frame Cloud Storage Servers (CSSs), we use signature verification to prevent malicious auditors from framing honest CSSs. Finally, security analysis proves that our proposal is secure in the random oracle model. Analysis and simulation results show that our proposal is more efficient than current state-of-the-art schemes.