Thermal-diffusional pulsation behaviors in planar as well as outwardly and inwardly propagating white dwarf (WD) carbon flames are systematically studied. In the 1D numerical simulation, the asymptotic degenerate equation of state and simplified one-step reaction rates for nuclear reactions are used to study the flame propagation and pulsation in WDs. The numerical critical Zel'dovich numbers of planar flames at different densities (rho - 2, 3, and 4. x. 10(7) g cm(-3)) and of spherical flames (with curvature c = -0.01, 0, 0.01, and 0.05) at a particular density (rho = 2 x 10(7) g cm-3) are presented. Flame front pulsation in different environmental densities and temperatures are obtained to form the regime diagram of pulsation, showing that carbon flames pulsate in the typical density of 2 x. 10(7) g cm(-3) and temperature of 0.6. x. 10(9) K. While being stable at higher temperatures, at relatively lower temperatures, the amplitude of the flame pulsation becomes larger. In outwardly propagating spherical flames the pulsation instability is enhanced and flames are also easier to quench due to pulsation at small radius, while the inwardly propagating flames are more stable.