Much biological experimental data are represented as curves, including measurements of growth, hormone, or enzyme levels, and physical structures. Here we consider the multiple testing problem of comparing two or more nonlinear curves. We model smooth curves of unknown form nonparametrically using penalized splines. We use random effects to model subject-specific deviations from the group-level curve. We present an approach that allows examination of overall differences between the curves of multiple groups and detection of sections in which the curves differ. Adjusted p-values for each single comparison can be obtained by exploiting the connection between semiparametric mixed models and linear mixed models and employing an approach for multiple testing in general parametric models. In simulations, we show that the probability of false-positive findings of differences between any two curves in at least one position can be controlled by a pre-specified error level. We apply our method to compare curves describing the form of the mouse dorsal funiculus - a morphological curved structure in the spinal cord - in mice wild-type for the gene encoding EphA4 or heterozygous with one of two mutations in the gene.