Negativity in a quasi-probability representation is typically interpreted as an indication of nonclassical behavior.  However, this does not preclude bases that are non-negative from having interesting applications---the single-qubit stabilizer states have non-negative Wigner functions and yet play a fundamental role in many quantum information tasks. We determine what other sets of quantum states and measurements of a qubit can be non-negative in a quasiprobability representation, and identify nontrivial groups of unitary transformations that permute such...