Aims. The purpose of this work is the characterization of the main scaling relations between all of the interstellar medium (ISM) components, namely dust, atomic, molecular, and total gas, and gas-phase metallicity, as well as other galaxy properties, such as stellar mass (M_star) and galaxy morphology, for late-type galaxies in the Local Universe. Methods. This study was performed by extracting late-type galaxies from the entire DustPedia sample and by exploiting the large and homogeneous dataset available thanks to the DustPedia project. The sample consists of 436 galaxies with morphological stage spanning from T = 1−10, M_star from 6 × 10⁷ to 3 × 10¹¹ M_⊙, star formation rate from 6 × 10⁻⁴ to 60 M_⊙ yr⁻¹, and oxygen abundance from 12 + log(O/H) = 8−9.5. Molecular and atomic gas data were collected from the literature and properly homogenized. All the masses involved in our analysis refer to the values within the optical disks of galaxies. The scaling relations involving the molecular gas are studied by assuming both a constant and a metallicity-dependent CO-to-H₂ conversion factor (X_CO). The analysis was performed by means of the survival analysis technique, in order to properly take into account the presence of both detection and nondetection in the data. Results. We confirm that the dust mass correlates very well with the total gas mass, and find –for the first time– that the dust mass correlates better with the atomic gas mass than with the molecular one. We characterize important mass ratios such as the gas fraction, the molecular-to-atomic gas mass ratio, the dust-to-total gas mass ratio (DGR), and the dust-to-stellar mass ratio, and study how they relate to each other, to galaxy morphology, and to gas-phase metallicity. Only the assumption of a metallicity-dependent X_CO reproduces the expected decrease of the DGR with increasing morphological stage and decreasing gas-phase metallicity, with a slope of about 1. The DGR, the gas-phase metallicity, and the dust-to-stellar mass ratio are, for our galaxy sample, directly linked to galaxy morphology. The molecular-to-atomic gas mass ratio and the DGR show a positive correlation for low molecular gas fractions, but for galaxies rich in molecular gas this trend breaks down. To our knowledge, this trend has never been found before, and provides new constraints for theoretical models of galaxy evolution and a reference for high-redshift studies. We discuss several scenarios related to this finding. Conclusions. The DustPedia database of late-type galaxies is an extraordinary tool for the study of the ISM scaling relations, thanks to its homogeneous collection of data for the different ISM components. The database is made publicly available to the whole community.