Numerical renormalization group simulations have shown that the underscreened spin-1 Kondo impurity model with power-law bath density of states (DOS) ρ(ω) ω r possesses various intermediate-coupling fixed points, including a stable non-Fermi-liquid phase. In this paper we discuss the corresponding universal low-energy theories, obtain thermodynamic quantities and critical exponents by renormalization group analysis together with suitable expansions, and compare our results with numerical data. Whereas the particle-hole symmetric critical point can be controlled at weak coupling using a simple generalization of the spin-1 2 model, we show that the stable non-Fermi-liquid fixed point must be accessed near strong coupling via a mapping onto an effective ferromagnetic Seff=1 2 model with singular bath DOS with exponent reff=-r<0. In addition, we consider the particle-hole asymmetric critical fixed point, for which we propose a universal field theory involving the crossing between doublet and triplet levels.