Although it is commonly accepted that most macroeconomic variables are non-stationary, it is often difficult to identify the source of the non-stationarity. 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 distinguish. The goal of this article is to extend the classical testing framework of I(1) versus I(0) + trends and/or breaks by considering a more general class of models under the null hypothesis: fractionally integrated (FI) processes. The asymptotic properties of the proposed tests are derived and it is shown that they are very well-behaved in finite samples. An illustration using US inflation data is also provided.