Carbon fusion reactions 12C(12C,p)23Na and 12C(12C,α)20Ne play a key role in the evolution of massive stars and in explosive scenarios such as type-Ia supernovae and super-bursts in binary stars. A direct determination of their cross sections is extremely challenging and discrepancies exist between different data sets in the literature. Here we report the results of a direct measurement performed at the CIRCE Tandem Accelerator Laboratory in Caserta (Italy), using ΔE−E detectors for unambiguous charge identification. Cross sections were measured in the energy range Ec.m.=2.51−4.36 MeV with energy steps between 10 and 25 keV in the centre of mass. To our knowledge these represent the finest energy steps to date. Results are presented in the form of partial and summed astrophysical S~-factors for individual proton- and α-particle channels. Branching ratios of individual proton- and α-particle groups were found to vary significantly with energy. Angular distributions, albeit limited to three angles, were also found to be non-isotropic, which could be a potential explanation for the discrepancies observed among different data sets. Further efforts are ongoing to extend measurements to lower energies.