Kinetic roughening of flux fronts penetrating in superconducting thin films are studied by means of a high resolution magneto-optic technique. The roughening exponent (α = 0.64) and growth exponent (β = 0.65) obtained from a dynamic scaling analysis of the initial stage of flux penetration and, at small length scales, are characteristic for a static disorder dominated nonlinear diffusion such as also observed in the directed percolated depinning model. At large length scale, α = 0.46 indicates a transition towards dynamic stochastic disorder, similar to the behavior of Kardar-Parisi-Zhang systems. There is a striking similarity with the behavior of combustion fronts in burning paper.