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As a model system for two-dimensional interface dynamics, we study the wetting front of a lipid membrane moving over a solid substrate that is structured with regularly spaced pinning centers. By analyzing the contour of the front, we derive the normal growth rate and the relaxation coefficient. Both exhibit a 1/t(1/2) time dependence. Moreover, the friction coefficient and the line tension can be determined. Randomly distributed pinning centers cause a fractal contour line, whereas on surfaces that are artificially roughened, self-affine contour lines are observed. The latter exhibit an anomalous roughness exponent of zeta = 0.81+/-0.05.