The design of secure and efficient user authentication protocols for multi-server environment is becoming a hot research topic in the cryptographic protocol community. Based on the widely accepted adversary model, this paper analyzes three representative, recently proposed user authentication schemes for multi-server environment. The paper reveals that:(1) Wan, et al.'s scheme is subject to offline password guessing attack as opposed to the authors' claim, and it also cannot provide user anonymity and forward secrecy; (2) Amin, et al.'s scheme cannot withstand offline password guessing attack, cannot preserve user anonymity and is vulnerable to two kinds of forward secrecy issues; (3) Reedy, et al.'s scheme cannot resist against user impersonation attack and offline password guessing attack, and also falls short of user un-traceability. The paper highlights three principles for designing more robust anonymous multi-factor authentication schemes:Public key principle, user anonymity principle and forward secrecy principle, explaining the essential reasons for the security flaws of the above protocols. It further proposes some amendments for the identified security flaws.