In this paper we use a general procedure to detect structural breaks at unknown points in time which allows for different orders of integration and deterministic components in each subsample (see Gil-Alana, 2006). First, we extend it to the non-linear case, and show by means of Monte Carlo experiments that the procedure performs well in a non-linear environment. Second, we apply it to test for breaks in the unemployment rate in the US, the UK and Japan. Our results shed some light on the empirical relevance of alternative unemployment theories for these countries. Specifically, a structuralist interpretation appears more appropriate for the US and Japan, whilst a hysteresis model accounts better for the UK experience (and also for the Japanese one in the second subsample). We interpret these findings in terms of different labour market features.