It is well known that fractionally integrated processes and short-memory models with trending components, possibly affected by structural breaks, imply similar features in the data and, accordingly, are hard to be distinguished. This paper proposes a state-space-model based test for the null hypothesis that a given time series, stationary and potentially long memory, is not affected by regime changes in the levels or by a smoothly varying trend. The state-space setup introduces a novel modeling approach in the long memory framework, which directly provides estimates of the probability and the size of the random level shifts, thus allowing to test for their joint nullity and to identify the break dates. A large set of Monte Carlo simulations shows that the proposed test, based on parametric estimates of the long memory parameter, is not distorted in finite samples and it has the highest power compared to other existing tests, based on semiparametric estimates of the long memory parameter