In Poland, special attention is focused on sustainable municipal waste management. As a result, new waste incineration plants are being planned. They are considered to be modern, ecologically friendly, and renewable energy sources. The waste from conventional incineration, which contains hazardous substances, must be disposed of in an appropriate manner. This study used advanced statistical tools, such as control charts, trend analysis, and time series analysis. The analysis was based on the leachability of selected elements and chemical compounds in incineration bottom ashes (IBAs) from the Waste to Energy Plant in Kraków, which were weathered for 2 weeks. The analysis was performed for 34 weeks. The obtained leachability results were compared with the leachability limit values of individual components. Based on the analysis of the control charts, it was found that in the case of selected samples, the leachability limit values for processing outside the plant using the R5 recovery process (LLVR5) values were exceeded. Seasonality analysis was performed using the autocorrelation function (ACF), the partial autocorrelation function (PACF), and the frequency analysis. Based on the obtained results, it was concluded that the leachability of elements and chemical compounds from waste does not confirm the occurrence of seasonality. It was found that from the exceedances of the LLVR5 mean that the two-week weathering is not sufficient and further studies should be carried out. The research methodology, which was presented on the example of the leachability of elements and compounds from IBA, can also be used for other waste analyses.