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 T(c)/2 to 2T(c), omitting a window of +/-1K about T(c), incorporate a three-peaks phonon spectrum and a mean-field jump. We find that a model including critical fluctuations with a critical exponent of almost-equal-to 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/T(c)-1\ > 0.085. The magnetic susceptibility above T(c) may be described equally well by critical fluctuations with an exponent of almost-equal-to 0.35 or by 2D/3D Gaussian fluctuations.