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In this work we present computer simulations of the dynamics of phase separation of a binary fluid on a variety of curved surfaces. The geometries considered range from sphere to ellipsoid, to models for budding cells, described as spheres with a bump. We study both the case in which there is no coupling between curvature and local composition, and the situation in which one of the two fluids, or lipids, prefers, or avoids, highly curved regions on the surface. Our results show that in these two cases there is a distinct difference in both the kinetic pathway to lipid sorting and the final localisation of the domains formed. Highlighting these differences should be useful, we expect, to discriminate between different theories for domain formation in a biophysical context.