After a brief review of current ideas on stripe order in cuprate high-temperature superconductors, we discuss the quasiparticle Nernst effect in cuprates, with focus on its evolution in non-superconducting stripe and related nematic states. In general, we find the Nernst signal to be strongly enhanced by nearby van-Hove singularities and Lifshitz transitions in the band structure, implying that phases with translation symmetry breaking often lead to a large quasiparticle Nernst effect due to the presence of multiple small Fermi pockets. Open orbits may contribute to the Nernst signal as well, but in a strongly anisotropic fashion. We discuss our results in the light of recent proposals for a specific Lifshitz transition in underdoped YBa₂Cu ₃O_y and make predictions for the doping dependence of the Nernst signal.