This work determines the critical regimes beyond which the melt fiber spinning for noncrystallizable polymeric liquids becomes unstable. The critical draw ratio of the process is established using linear stability analysis for both isothermal and nonisothermal fiber spinning regimes. In addition, nonlinear isothermal analysis describes the complete range of the stable and unstable conditions for fiber spinning. Unlike previous studies, this research uses quite realistic viscoelastic constitutive equations extensively tested for five polymer liquids, which provides a good comparison of our calculations with available experimental data. POLYM. ENG. SCI., 50:1414-1423, 2010. (C) 2010 Society of Plastics Engineers