  In this work, we present a new catalogue of >40000 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 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.
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), 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 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 DIG-corrected line fluxes prompts the use
of a higher theoretical Ha/Hb ratio (3.03) than what is commonly used when
recovering the E(B-V) via the Balmer decrement technique in massive
star-forming galaxies.
