We consider experimentally the viscous fingering problem wherein a silicone oil displaces a Carbopol gel in a rectangular Hele-Shaw cell. Unlike traditional cases, the displacing fluid perfectly wets the walls of the Hele-Shaw cell. While the silicone oil used is a Newtonian fluid (with a small viscosity), the Carbopol gel employed is a non-Newtonian fluid (with a large viscosity), which exhibits an interesting range of complex rheology such as yield stress, shear-thinning and weakly-elastic behaviours. We demonstrate that non-Newtonian effects together with wettability conditions exert a significant impact on fingering patterns, which we describe through a detailed comparison between wetting displacements (oil-Carbopol) and non-wetting displacements (air-Carbopol). We analyze the flow from various perspectives, using the relevant dimensionless groups, including the capillary number (Ca), the Bingham number (En), the Weissenberg number (Wi), Weber number (We), the channel aspect ratio (delta), and the shear-thinning power-law index (n). We succeed in classifying four main flow regimes versus these dimensionless groups: a capillary regime, a yield stress regime, a viscous regime and an elasto-inertial regime. The capillary regime, observed at very low velocities, is a unique regime which appears due to an interplay between yield stress and interfacial tension effects. In this regime, the finger width is nearly equal to the cell width. The other flow regimes are observed in both wetting and non-wetting displacements, but with certain critical differences that are made clear in the paper.