In this paper we use the constitutive equation of Bhave et al. (1993) for rod‐like, liquid‐crystalline polymer solutions to analyze the isothermal, steady‐state spinning of these liquids in order to understand the evolution of microstructure, predict the velocity and normal stress distributions in the filament, and examine the effect of different upstream microstructural conditions. Our analysis shows that in contrast to fiber spinning models of isotropic liquids, the velocity, structure, and stress profiles are sensitive to the choice of initial conditions. In addition we have investigated the impact of the closure approximation used in the constitutive equation of Bhave et al. on the fiber spinning problem by solving the equation for the distribution function directly; only slight changes are seen in the velocity and stress profiles. An apparent elongational viscosity defined as the ratio of normal stress difference to strain rate at the takeup compares very well with the true elongational viscosity η̄ for the model, thereby suggesting that fiber spinning flows can be used to determine η̄ for liquid‐crystalline polymer solutions. Model predictions of the velocity and stress agree well with data obtained by Prilutski (1984) for HPC/acetic acid solutions. Finally, we present a linear stability analysis of the spinning problem to show the impact of viscoelasticity, inertia, gravity, and surface tension on the onset of draw resonance instabilities. The neutral stability curves obtained for dominant viscoelastic forces reflect trends in the apparent elongational viscosity. Model predictions are in qualitative agreement with the draw resonance data reported by Prilutski.