We review data analysis techniques that can be used to study temporal correlations among conductance traces in break junction measurements. We show that temporal histograms are a simple but efficient tool to check the temporal homogeneity of the conductance traces, or to follow spontaneous or triggered temporal variations, like structural modifications in trained contacts, or the emergence of single-molecule signatures after molecule dosing. To statistically analyze the presence and the decay time of temporal correlations, we introduce shifted correlation plots. Finally, we demonstrate that correlations between the opening and subsequent closing traces may indicate structural memory effects in atomic-sized metallic and molecular junctions. Applying these methods on measured and simulated gold metallic contacts as a test system, we show that the surface diffusion induced flattening of the broken junctions helps to produce statistically independent conductance traces at room temperature, whereas at low temperature repeating tendencies are observed as long as the contacts are not closed to sufficiently high conductance setpoints. Applying opening-closing correlation analysis on Pt-CO-Pt single-molecule junctions, we demonstrate pronounced contact memory effects and recovery of the molecule for junctions breaking before atomic chains are formed. However, if chains are pulled the random relaxation of the chain and molecule after rupture prevents opening-closing correlations.