||In this work, we present a new catalogue of >40 000 ionised nebulae distributed across the 19 galaxies observed by the PHANGS-MUSE survey. The nebulae have been classified using a new model-comparison-based algorithm that exploits the odds ratio principle to assign a probabilistic classification to each nebula in the sample. The resulting catalogue is the largest catalogue containing complete spectral and spatial information for a variety of ionised nebulae available so far in the literature. We developed this new algorithm to address some of the main limitations of the traditional classification criteria, such as their binarity, the sharpness of the involved limits, and the limited amount of data they rely on for the classification. The analysis of the catalogue shows that the algorithm performs well when selecting H II regions. In fact, we can recover their luminosity function, and its properties are in line with what is available in the literature. We also identify a rather significant population of shock-ionised regions (mostly composed of supernova remnants), which is an order of magnitude larger than any other homogeneous catalogue of supernova remnants currently available in the literature. The number of supernova remnants we identify per galaxy is in line with results in our Galaxy and in other very nearby sources. However, limitations in the source detection algorithm result in an incomplete sample of planetary nebulae, even though their classification seems robust. Finally, we demonstrate how applying a correction for the contribution of the diffuse ionised gas to the nebulae’s spectra is essential to obtain a robust classification of the objects and how a correct measurement of the extinction using diffuse-ionised-gas-corrected line fluxes prompts the use of a higher theoretical Hα/Hβ ratio (3.03) than what is commonly used when recovering the E(B – V) via the Balmer decrement technique in massive star-forming galaxies.