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 the depth of the observation (where a 'good' detection is taken to be one with positional offset less than one full-width half-maximum of the point-response function, and with the measured flux density within a factor of 2 of the flux density of the injected source). This paper acts as a reference for the 2010 July HerMES public data release. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.