Quantiles provide a comprehensive description of the properties of a variable, and tracking changes in quantiles over time using signal extraction methods can be informative. It is shown here how departures from strict stationarity can be detected using stationarity tests based on weighted quantile indicators. Corresponding tests based on expectiles are also proposed; these might be expected to be more powerful for distributions that are not heavy-tailed. Tests for changing dispersion and asymmetry may be based on contrasts between particular quantiles or expectiles. An overall test of the null hypothesis of strict stationarity can be constructed using the indicators from a range of quantiles. Residuals from fitting a time-varying level or trend may be used to construct tests for relative time invariance. Empirical examples, using stock returns and US inflation, demonstrate the practical value of the tests.