Eutectic dendrites forming in a model ternary system have been studied using the phase-field theory. The eutectic and one-phase dendrites have similar forms, and the tip radius scales with the interface free energy as for one-phase dendrites. The steady-state eutectic patterns appearing on these two-phase dendrites include concentric rings, and single- to multiarm spirals, of which the fluctuations choose, a stochastic phenomenon characterized by a peaked probability distribution. The number of spiral arms correlates with tip radius and the kinetic anisotropy.