An experimental investigation of the flow-front behavior of dilute and semi-dilute polymer solutions has been carried out to gain a better understanding of the underlying mechanisms leading to the occurrence of an unstable flow at the advancing flow-front during the filling of a rectangular Hele-Shaw cell. Our experimental results have revealed the existence of an elastic finger-like instability at the advancing flow-front that develops in semi-dilute solutions of high molecular weight polymers, with an onset time of approximately a few hundred milliseconds. Although at shear rates above critical, narrow finger patterns develop at the flow-front, their amplitude and number remain roughly constant throughout the flowing. At critical condition, no secondary flow was observed in the vicinity of the front region where the unstable flow develops. Transient response of the normal stress difference and the shear stress in the plate-and-plate geometry at shear rate above critical (for the elastic fingering instability in the Hele-Shaw cell) did not reveal any anomalous that could lead to the formation of such finger-like instabilities. These instabilities were observed for both the ideal elastic Boger fluids and shear thinning viscoelastic fluids.