Liquefying hybrid rocket fuels (e.g. paraffin) enable higher regression rates due to the presence of an unstable melt layer on the fuel surface during combustion, which causes entrainment of liquid droplets into the oxidizer gas flow. In order to better understand the mechanism responsible for the droplets entrainment, the combustion behaviour of paraffin-based hybrid rocket fuels in combination with gaseous oxygen (GOX) was investigated in the framework of this research. Tests were performed in a 2D slab burner configuration at atmospheric conditions. High-speed videos were recorded and analysed with two different decomposition techniques, applied to the scalar field of the flame luminosity (the flame front is assumed to follow the liquid layer). The fuel slab composition and configuration and the oxidizer mass flow have been varied in order to study the influence of these parameters on the liquid layer instability process. The main focus of the research is to understand the relation between the unstable waves which enable the droplets entrainment process and the regression rate. The results show that the combustion is dominated by periodic, wave-like structures for all the analysed fuels. The frequencies and the wavelengths characterizing the liquid melt layer depend on the fuel viscosity and geometry and on the oxidizer mass flow. Moreover, a dependency of the regression rate on the most excited frequencies and longitudinal wavelengths was found. This is important to better understand the relation between the increased regression rate and the onset and development of the entrainment process, which is connected to the amplification of longitudinal unstable waves caused by the high velocity gas flow over the fuel surface.