This paper reports on the geometric optimisation of a three-dimensional micro-channel heat sink. Two types of micro-channel heat sink configurations were studied. In both cases, the objective was to maximise the global thermal conductance subject to a fixed volume, high conducting material and a fixed pressure drop. In the first configuration, the micro-channel was completely embedded inside a high conducting material and numerical simulations were carried out on a unit cell with volume ranging from 0.1 to 0.9 mm³ and pressure drop between 10 and 75 kPa. The axial length of the micro-channel heat sink was fixed at 10 mm. The cross-sectional area of the micro-channel heat sink was free to morph with respect to the degrees of freedom provided by the aspect ratio and the solid volume fraction. The effects of the total solid volume fraction and the pressure drop on the aspect ratio, channel hydraulic diameter and peak temperature were investigated. In the second configuration the micro-channels were embedded inside a high conducting material, except that the top was covered with an insulating material. The whole configuration was allowed to morph with respect to all the degrees of freedom. Similar but dimensionless numerical simulations were carried out on this configuration and the numerical optimisation results were reported. In the first configuration numerical results show that the degrees of freedom have a strong effect on the peak temperature and the maximum thermal conductance. The optimal geometric characteristics (aspect ratio and the optimal channel shape) are reported and compared with those obtained from approximate relationships using scale analysis. For this configuration, the predicted trends are found to be in good agreement with the predicted results. In the second configuration, a test case on an actual micro-channel heat sink shows a reduction of about 8% in global thermal resistance.