In order to investigate the nature of the superconducting transition of YBa2Cu3O7, polycrystalline samples with calorimetric transition widths of 0.4 to 0.5 K were selected. Their specific heat was measured with 0.01% resolution and 0.5% accuracy. The models used to analyze the data from Tc/2 to 2Tc, omitting a window of ±1K about Tc, incorporate a three-peaks phonon spectrum and a mean-field jump. We find that a model including critical fluctuations with a critical exponent of ≈0.1 fits the data to 0.014% rms, while a model featuring 2D/3D Gaussian fluctuations fits “only” to 0.047% in the same conditions. The latter model becomes however acceptable for |T/Tc−1| > 0.085. The magnetic susceptibility above Tc may be described equally well by critical fluctuations with an exponent of ≈0.35 or by 2D/3D Gaussian fluctuations.