We introduce a general formalism combining the coupled oscillator model with the transfer matrix method to analyze and engineer the phase of the light reflected from a Fano-resonant metasurface. This method accounts for periodicity and the presence of substrates, and we demonstrate that these factors can be used to tune the reflected phase at will. Utilizing these effects and adjusting the coupling strength of the underlying unit cell, we achieve zero reflection at the dark resonance of the metasurface. We show that the resulting phase singularity can dramatically increase the sensitivity of phase-based detection schemes. The phase bifurcation unveiled in this work can be used to design plasmonic metasurfaces that explore the unusual phase behavior of light.