A very sensitive microcalorimeter has been recently built in order to measure the specific heat of bulk single crystals with a mass of order some μ g. The resolution achieved is 10-4 for the specific heat and 10-5 for the temperature. A systematic study of several twinned and untwinned single crystals of YBCO in zero field reveals that the behavior close to the superconducting transition is very sensitive to the degree of disorder in the sample. Gaussian fluctuations are observed in a large window around T_c, and critical fluctuations are possibly observed only 100mK around T_c. The role of twins in the cut-off of the divergence will be discussed. Preliminary data on the behavior in high magnetic fields will also be presented.