Bipartite all-versus-nothing proofs of Bell's theorem with single-qubit measurements

If we distribute n qubits between two parties, which quantum pure states and distributions of qubits would allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell's theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs for any number of qubits, and provide all distinct proofs up to n=7 qubits. Remarkably, there is only one distribution of a state of n=4 qubits, and six distributions, each for a different state of n=6 qubits, which allow these proofs.