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 Multi Unit Spectroscopic Explorer (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 (Physics at High Angular resolution in Nearby GalaxieS)-MUSE project, we identify SNRs in 19 nearby galaxies at ~100pc scales. We use five different optical diagnostics: (1) line ratio maps of [SII]/H{alpha}; (2) line ratio maps of [OI]/H{alpha}; (3) velocity dispersion map of the gas; (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 HII regions and diffuse ionized gas, in our line ratio maps we use a novel technique to search for objects with [SII]/H{alpha} or [OI]/H{alpha} in excess of what is expected at fixed H{alpha} surface brightness within photoionized gas. In total, we identify 2233 objects using at least one of our diagnostics, and define a subsample of 1,166 high-confidence SNRs that have been 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 SNRs we found are also identified in literature catalogs and we recover 51% of the known SNRs. The remaining 1067 objects in our sample are detected with only one diagnostic and we classify them as SNR candidates. We find that ~35% of all our objects overlap with the boundaries of HII regions from literature catalogs, highlighting the importance of using indicators beyond line intensity morphology to select SNRs. We find that the [OI]/H{alpha} line ratio is responsible for selecting the most objects (1368; 61%), however, only half are classified as SNRs, demonstrating how the use of multiple diagnostics is key to both increasing our sample size and improving our confidence in our SNR classifications.