The growing number of variable speed drives (VSDs) in industry has an impact on the future development of condition monitoring methods. In research, more and more attention is being paid to condition monitoring based on motor current evaluation. However, there are currently only a few contributions to current-based pump diagnosis. In this paper, two current-based methods for the detection of bearing defects, impeller clogging, and cracked impellers are presented. The first approach, load point-dependent fault indicator analysis (LoPoFIA), is an approach that was derived from motor current signature analysis (MCSA). Compared to MCSA, the novelty of LoPoFIA is that only amplitudes at typical fault frequencies in the current spectrum are considered as a function of the hydraulic load point. The second approach is advanced transient current signature analysis (ATCSA), which represents a time-frequency analysis of a current signal during start-up. According to the literature, ATCSA is mainly used for motor diagnosis. As a test item, a VSD-driven circulation pump was measured in a pump test bench. Compared to MCSA, both LoPoFIA and ATCSA showed improvements in terms of minimizing false alarms. However, LoPoFIA simplifies the separation of bearing defects and impeller defects, as impeller defects especially influence higher flow ranges. Compared to LoPoFIA, ATCSA represents a more efficient method in terms of minimizing measurement effort. In summary, both LoPoFIA and ATCSA provide important insights into the behavior of faulty pumps and can be advantageous compared to MCSA in terms of false alarms and fault separation.