The time series recordings of typical songs of songbirds exhibit highly complex and structured behavior, which is characteristic of their species and stage of development, and need to be analyzed by methods that can uncover their correlation structure. Here we analyze a typical song of a canary using Hurst exponents and multifractal analysis, which uncovers the correlation structure of typical song segments. These are then compared with the corresponding quantities from shuffled data, which destroys the temporal correlations and iterative amplitude-adjusted Fourier transform (IAAFT) data. It is seen that temporal correlations are responsible for the multifractal behavior seen in the data and that two-point correlations, which are preserved by the transform, are important in the high-fluctuation regime. Higher-order correlations and intersyllabic gaps dominate the behavior of the low-fluctuation regime. These observations are supported by the simplicial characterization of the corresponding time series networks. Complexity measures are also used to analyze the amplitude envelope time series. These indicate that intersyllabic gaps contribute a significant fraction to the complexity of the birdsong. Our method provides a detailed characterization of the data, which can enable the comparison of real and synthetic birdsong and comparisons across stages of development and species. A brief comparison with the song of the zebra finch supports this.