We describe the generation of single-band point source catalogues from submillimetre Herschel-SPIRE observations taken as part of the Science Demonstration Phase of the Herschel Multi-tiered Extragalactic Survey (HerMES). Flux densities are found by means of peak finding and the fitting of a Gaussian point-response function. With highly confused images, careful checks must be made on the completeness and flux-density accuracy of the detected sources. This is done by injecting artificial sources into the images and analysing the resulting catalogues. Measured flux densities at which 50 per cent of injected sources result in good detections at (250, 350 and 500) μm range from (11.6, 13.2 and 13.1) to (25.7, 27.1 and 35.8) mJy, depending on t...