We have studied in detail a sample of 967 SPIRE sources with 5 sigma detections at 350 and 500 p.m and associations with Spitzer-S WIRE 24 p.m galaxies in the HerMES-Lockman survey area, fitting their mid- and far-infrared, and submillimetre, spectral energy distributions (SEDs) in an automatic search with a set of six infrared templates. For almost 300 galaxies, we have modelled their SEDs individually to ensure the physicality of the fits. We confirm the need for the new cool and cold cirrus templates, and also of the young starburst template, introduced in earlier work. We also identify 109 lensing candidates via their anomalous SEDs and provide a set of colour redshift constraints which allow lensing candidates to be identified from combined Herschel and Spitzer data. The picture that emerges of the submillimetre galaxy population is complex, comprising ultraluminous and hyperluminous starbursts, lower luminosity galaxies dominated by interstellar dust emission, lensed galaxies and galaxies with surprisingly cold (10-13 K) dust. 11 per cent of 500 p.m selected sources are lensing candidates. 70 per cent of the unlensed sources are ultraluminous infrared galaxies and 26 per cent are hyperluminous. 34 per cent are dominated by optically thin interstellar dust ('cirrus') emission, but most of these are due to cooler dust than is characteristic of our Galaxy. At the highest infrared luminosities we see SEDs dominated by M82, Arp 220 and young starburst types, in roughly equal proportions.