Intrapartum fetal heart rate (FHR), routinely monitored in daily obstetrical practice, enables early detection of fetal asphyxia and thus prevention of labour adverse outcomes. FHR variability (FHRV) constitutes an essential characterization of fetal well-being; fetuses with large FHRV are unlikely to be at risk of brain injury. This study investigates the impacts of labour first and second stages on the characterization of FHR temporal dynamics as well as on the discrimination of healthy from acidotic fetuses. FHR temporal dynamics are quantified using Hurst parameter, H, practically estimated within a wavelet framework. Analyses are performed on first and second stages, over a large (3049 records) and well documented database, collected at Hôpital Femme-Mère-Enfant, in Lyon, France. It is observed that Ĥ, for healthy fetuses, remains constant along first stage and then significantly increases during second stage; while, for acidotic fetuses, Ĥ increases to larger values earlier within first stage. This may indicate that the defense mechanism classically at work during second stage as a reaction to excessive stress, is already in use during first stage for acidotic fetuses. Detection performance curves (ROC) also show that second stage Ĥ permits slightly better discrimination of healthy from acidotic fetuses, compared to first stage Ĥ.