We explore the interference of two bichromatic vector beams in Young's interference experiment. Our analysis focuses on determining the conditions under which the superposition of such beams, emerging from the pinholes, can give rise to Lissajous-type polarization singularities on the observation screen. Two independent sufficiency conditions are derived. This analysis aids in comprehending the inherent characteristics of Lissajous singularities. To the best of our knowledge, this is the first demonstration of the singular behavior of polarization in a two-frequency field in Young's interference experiment.