We discuss the two-channel Kondo problem with a pseudogap density of states p(ω) α |ω|^r of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic r = 0 case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for 0 < r < r_max ≈ 0.23. For r > r_max, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively.We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ ϵ expansion techniques to calculate critical properties near the critical dimensions r = 0 and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene.We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.