A number of morphological and statistical aspects of domain formation in singly and doubly supported ternary membranes have been investigated. Such ternary membranes produce macroscopic phase separation in two fluid phases and are widely used as raft models. We find that membrane interactions with the support surface can have a critical influence on the domain shapes if measures are not taken to screen these interactions. Combined AFM and fluorescence microscopy demonstrate small (500 nm) irregular domains and incomplete formation of much larger (5 mu m) round domains. These kinetically trapped structures are the result of interactions between the membrane and the support surface, and they can be effectively removed by employing doubly supported membranes under physiological salt concentrations. These decoupled supported membranes display macroscopic round domains that are easily perturbed by fluid shear flow. The system allows a quantitative characterization of domain coarsening upon being cooled into the coexistence region. We determine the domain growth exponent alpha = 0.31, which is in close agrement with the theoretical value of 1/3. Analysis of the spatial domain pattern in terms of Voronoi polygons demonstrates a close similarity to equilibrated cellular structures with a maximized configurational entropy.