Intrapartum fetal heart rate monitoring, aiming at early acidosis detection, constitutes an important public health stake. Scattering transform is proposed here as a new tool to analyze intrapartum fetal heart rate (FHR) variability. It consists of a nonlinear extension of the underlying wavelet transform, that thus preserves its multiscale nature. Applied to an FHR signal database constructed in a French academic hospital, the scattering transform is shown to permit to efficiently measure scaling exponents characterizing the fractal properties of intrapartum FHR temporal dynamics, that relate not only to the sole covariance (correlation scaling exponent), but also to the full dependence structure of data (intermittency scaling exponent). Such exponents are found to satisfactorily discriminate temporal dynamics of healthy subjects (from that of nonhealthy ones) and to emphasize the role of the highest frequencies (around and above 1 Hz) in intrapartum FHR variability. This permits us to achieve satisfactory classification performance that improves on those obtained from the analysis of International Federation of Gynecology and Obstetrics (FIGO) criteria, notably by classifying as healthy a number of subjects that were incorrectly classified as nonhealthy by classical clinically used FIGO criteria. Combined to obstetrician annotations, these scaling exponents enable us to sketch a typology of these FIGO-false positive subjects. Also, they permit us to monitor the evolution along time of the intrapartum health status of the fetuses and to estimate an optimal detection time-frame.