Oxy-combustion characteristics of methane and hydrogen-enriched methane have been investigated experimentally in a nonpremixed swirl-stabilized combustor. Experiments were conducted at different firing rates ranging from 2.S to 4.5 MW/m(3)-bar and 0% to 20% hydrogen content in methane/hydrogen fuel mixtures. When the combustor is operated under gas turbine conditions (>= 3.5 MW/m(3)-bar), the flame transitions exhibit trimodal regime (i.e., attached flame -> lifted flame -> no flame) below which the flame transitions exhibit bimodal regime (i.e., attached flame -> no flame). Weak flames at the nozzle exit were generally observed to precede the attached flame -> lifted flame transition. The weak flame is due to the entrainment of more CO2-containing oxidizer (O-2/CO2) to the fuel stream that reduces the flame burning rate. The attached flame -> lifted flame transition occurs at a critical oxidizer velocity. The critical oxidizer velocity strongly depends on the fuel composition. Moreover, lifted flame oscillates about different points (stabilization points) within the combustor. These points can be interpreted as locations of lower scalar dissipation rate, where the leading edge flame speed matches the local flow velocity. Empirical equations presented in this study captured the trend of our experimentally normalized flame length. The predicted flame length based on the near-field concept gave a good match with our experimentally observed flame length. Temperature data are also presented and can be used in the validation of numerical models to have further insight into the oxy-combustion dynamics of methane and hydrogen-enriched methane in a cost-effective way.