We investigate the stability of theories in which Lorentz invariance is spontaneously broken by fixed-norm vector "aether" fields. Models with generic kinetic terms are plagued either by ghosts or by tachyons, and are therefore physically unacceptable. There are precisely three kinetic terms that are not manifestly unstable: a sigma model $(\partial_μA_ν)2$, the Maxwell Lagrangian $F_{μν}F^{μν}$, and a scalar Lagrangian $(\partial_μA^μ)2$. The timelike sigma-model case is well-defined and stable when the vector norm is fixed by a constraint; however, when it is determined by minimizing a potential there is necessarily a tachyonic ghost, and therefore an instability. In the Maxwell and scalar cases, the Hamiltonian is unbounded below, but at the level of perturbation theory there are fewer degrees of freedom and the models are stable. However, in these two theories there are obstacles to smooth evolution for certain choices of initial data.

23 pages, 1 figure; in v2: added references and made minor minor changes to text in sections I and VI; in v3 (version accepted by PRD): added references and made minor changes to discussion in section III