Supernova feedback injects energy and turbulence into the interstellar medium (ISM) in galaxies, influences the process of star formation,
and is essential to understanding the formation and evolution of galaxies. In this paper we present the largest extragalactic survey of supernova remnant (SNR) candidates in nearby star-forming galaxies using exquisite spectroscopic maps from MUSE. Supernova remnants (SNRs) exhibit distinctive emission-line ratios and kinematic signatures, which are apparent in optical spectroscopy. Using optical integral field spectra from the PHANGS--MUSE project, we identified SNRs in 19 nearby galaxies at sim 100 pc scales. We used five different optical diagnostics: (1) line ratio maps of S ii /Halpha ; (2) line ratio maps of O i /Halpha ; (3) velocity dispersion map of the gas; and (4) and (5) two line ratio diagnostic diagrams from Baldwin, Phillips Terlevich (BPT) diagrams to identify and distinguish SNRs from other nebulae. Given that our SNRs are seen in projection against H ii regions and diffuse ionized gas, in our line ratio maps we used a novel technique to search for objects with S ii /Halpha or O i /Halpha in excess of what is expected at fixed Halpha surface brightness within photoionized gas. In total, we identified 2,233 objects using at least one of our diagnostics, and defined a subsample of 1,166 high-confidence SNRs that were detected with at least two diagnostics. The line ratios of these SNRs agree well with the MAPPINGS shock models and we validate our technique using the well-studied nearby galaxy M83, where all the SNRs we found are also identified in literature catalogs, and we recovered 51<!PCT!> of the known SNRs. The remaining 1,067 objects in our sample were detected with only one diagnostic, and we classified them as SNR candidates. We find that sim 35<!PCT!> of all our objects overlap with the boundaries of H ii regions from literature catalogs, highlighting the importance of using indicators beyond line intensity morphology to select SNRs We find that the O i /Halpha line ratio is responsible for selecting the most objects (1,368; 61<!PCT!>); however, only half are classified as SNRs, demonstrating how the use of multiple diagnostics is key to increasing our sample size and improving our confidence in our SNR classifications.