We consider fiber spinning for the upper-convected Maxwell fluid in the limit of a high Deborah number. We compare several choices of boundary conditions that may be imposed. In addition to the takeup speed and the upstream flow rate, we consider four different boundary conditions: the upstream velocity, upstream elastic stress, the force in the fiber, and the ratio of stress to the square of the velocity (the latter can be motivated by a limit of vanishing retardation time). We find that the effect of the boundary condition on stability is crucial; in one case we even find an instability even though the draw ratio is 1.