The phase-shifting interferometry has been intensively studied for more than half a century, and is still actively investigated and improved for more demanding precision measurement requirements. A proper phase-shifting algorithm (PSA) for phase extraction should consider various error sources including (i) the phase-shift errors, (ii) the intensity harmonics, (iii) the non-uniform phase-shift distributions and (iv) the random additive intensity noise. Consequently, a large pool of PSAs has been developed, including those with known phase shifts (abbreviated as kPSA) and those with unknown phase shifts (abbreviated as uPSA). While numerous evaluation works have been done for the kPSAs, there are very few for the uPSAs, making the overall picture of the PSAs unclear. Specifically, there is a lack of (i) fringe pattern parameters’ restriction analysis for the uPSAs and (ii) performance comparison within the uPSAs and between the uPSAs and the kPSAs. Thus, for the first time, we comprehensively evaluated the pre-requisites and performance of four representative uPSAs, the advanced iterative algorithm, the general iterative algorithm (GIA), the algorithm based on the principal component analysis and the algorithm based on VU factorization, and then compare the uPSAs with twelve benchmarking kPSAs. From this comparison, the demand for proper selection of a kPSA, and the restriction and attractive performance of the uPSAs are clearly depicted. Due to the outstanding performance of the GIA, a hybrid kPSA-GIA is proposed to boost the performance of a kPSA and relieve the fringe density restriction of the GIA.