The paper deals with the regular refraction of a plane shock at a gas interface for the particular case where the reflected wave is an expansion fan. Numerical results are presented for the air–CH4 and air–CO2 gas combinations which are respectively examples of ‘slow–fast’ and ‘fast–slow’ refractions. It is found that a previously unreported condition exists in which the reflected wave solutions may be multi-valued. The hodograph mapping theory predicts a new type of regular–irregular transition for a refraction in this condition. The continuous expansion wave type of irregular refraction is also examined. The existence of this wave system is found to depend on the flow being self-similar. By contrast the expansion wave becomes centred when the flow becomes steady. Transitions within the ordered set of regular solutions are examined and it is shown that they may be either continuous or discontinuous. The continuous types appear to be associated with fixed boundaries and the discontinuous types with movable boundaries. Finally, a number of almost linear relations between the wave strengths are noted.