The large range of time and length scales involved in Type Ia supernovae (SNe Ia) requires the use of flame models. As a prelude to exploring various options for flame models, we consider in this paper high-resolution, three-dimensional simulations of the small-scale dynamics of nuclear flames in the supernova environment in which the details of the flame structure are fully resolved. The range of densities examined, (1–8) × 10⁷ g cm⁻³, spans the transition from the laminar flamelet regime to the distributed burning regime where small-scale turbulence disrupts the flame. The use of a low Mach number algorithm facilitates the accurate resolution of the thermal structure of the flame and the inviscid turbulent kinetic energy cascade, while implicitly incorporating kinetic energy dissipation at the grid-scale cutoff. For an assumed background of isotropic Kolmogorov turbulence with an energy characteristic of SNe Ia, we find a transition density between 1 and 3 × 10⁷ g cm⁻³, where the nature of the burning changes qualitatively. By 1 × 10⁷ g cm⁻³, energy diffusion by conduction and radiation is exceeded, on the flame scale, by turbulent advection. As a result, the effective Lewis number approaches unity. That is, the flame resembles a laminar flame but is turbulently broadened with an effective diffusion coefficient, D_T ~ u^'l, where u^' is the turbulent intensity and l is the integral scale. For the larger integral scales characteristic of a real supernova, the flame structure is predicted to become complex and unsteady. Implications for a possible transition to detonation are discussed.