First-order separability of a spatio-temporal point process plays a fundamental role in theanalysis of spatio-temporal point pattern data. While it is often a convenient assumptionthat simplifies the analysis greatly, existing non-separable structures should be accountedfor in the model construction. Three different tests are proposed to investigate thishypothesis as a step of preliminary data analysis. The first two tests are exact orasymptotically exact for Poisson processes. The first test based on permutations and globalenvelopes allows one to detect at which spatial and temporal locations or lags the datadeviate from the null hypothesis. The second test is a simple and computationally cheapχ2-test. The third test is based on stochastic reconstruction method and can be generallyapplied for non-Poisson processes. The performance of the first two tests is studied in asimulation study for Poisson and non-Poisson models. The third test is applied to the realdata of the UK 2001 epidemic foot and mouth disease.