Model membrane systems are a useful tool for studying the role of lateral phase separation in relation to the lipid rafts on living cells. One key aspect of the phase separation observed in both living and model systems is phase morphology, the size and shape of phase domains. For many lipid systems the phase morphology tends towards a single large round (macroscopic) domain to minimize the perimeter to area ratio. By contrast, several lipid systems have been found to have nanoscopic (nanodomains) or modulated (periodic and thermodynamically stable) phase domains. The explanation for the large excess in phase boundary present in these mixtures requires the use of a competing interactions model, in which line tension (energy per unit length) competes with curvature and/or electrostatics to stabilize non-trivial phase morphologies. To study these interactions in a way that accurately represents the membrane shape we present and implement a lattice model for use in computer simulations of phase morphology. These simulations show that on a spherical surface, curvature can compete with line tension to produce modulated phases that closely match the size and characteristics observed on giant unilamellar vesicles (GUV). The model is extended to include electrostatic repulsion, which is found to break up macroscopic domains on large unilamellar vesicles (LUV) into irregular clusters with correlation lengths consistent with size measurements of nanodomains.