This contribution describes measurements of lipid bilayer domain line tension based on two-dimensional thermal undulations of membranes with liquid ordered/liquid disordered phase coexistence and near-critical composition at room temperature. Lateral inhomogeneity of lipid and protein composition is currently a subject of avid research aimed at determining both fundamental properties and biological relevance of membrane domains. Line tension at fluid lipid bilayer membrane domain boundaries controls the kinetics of domain growth and therefore regulates the size of compositional heterogeneities. High line tension promotes membrane domain budding and fission. Line tension could therefore be an important control parameter regulating functional aspects of biological membranes. Here the established method of fluid domain flicker spectroscopy is applied to examine thermal domain wall fluctuations of phase-separated bilayer membranes. We find a Gaussian probability distribution for the first few excited mode amplitudes, which permits an analysis by means of appropriately specialized capillary wave theory. Time autocorrelation functions are found to decay exponentially, and relaxation times are fitted by means of a hydrodynamic theory relating line tensions and excited mode relaxation kinetics. Line tensions below 1 pN are obtained, with these two approaches yielding similar results. We examine experimental artifacts that perturb the Fourier spectrum of domain traces and discuss ways to identify the number of modes that yield reliable line tension information.