The power law process (PLP) is often used to model failure data from repairable systems, and its adequacy is tested with goodness-of-fit tests such as the Cramér–von Mises test. No single test, however, can be sensitive to all possible deviations from the PLP. Exact tests that are powerful against some departures from the PLP of particular interest to practitioners are derived heuristically, the computation of theirpvalues is described, and their use is illustrated with examples. One test is a modified version of the Laplace test for trend, another a modification of Greenwood's test for clustering. It is shown that these tests can be derived as score tests, which ensures that they are powerful for large samples.