We used wide-area surveys over 39 deg2 by the HerMES (Herschel Multi-tiered Extragalactic Survey) collaboration, performed with the Herschel Observatory SPIRE multiwavelength camera, to estimate the low-redshift, 0.02 < z < 0.5, monochromatic luminosity functions (LFs) of galaxies at 250, 350 and 500 μm. Within this redshift interval, we detected 7087 sources in five independent sky areas, ∼40 per cent of which have spectroscopic redshifts, while for the remaining objects photometric redshifts were used. The SPIRE LFs in different fields did not show any field-to-field variations beyond the small differences to be expected from cosmic variance. SPIRE flux densities were also combined with Spitzer photometry and multiwavelength archival data to perform a complete spectral energy distribution fitting analysis of SPIRE detected sources to calculate precise k-corrections, as well as the bolometric infrared (IR; 8–1000 μm) LFs and their low-z evolution from a combination of statistical estimators. Integration of the latter prompted us to also compute the local luminosity density and the comoving star formation rate density (SFRD) for our sources, and to compare them with theoretical predictions of galaxy formation models. The LFs show significant and rapid luminosity evolution already at low redshifts, 0.02 < z < 0.2, with L∗IR∝(1+z)6.0±0.4IR∗∝(1+z)6.0±0.4 and Φ∗IR∝(1+z)−2.1±0.4ΦIR∗∝(1+z)−2.1±0.4, L∗250∝(1+z)5.3±0.2250∗∝(1+z)5.3±0.2 and Φ∗250∝(1+z)−0.6±0.4Φ250∗∝(1+z)−0.6±0.4 estimated using the IR bolometric and the 250 μm LFs, respectively. Converting our IR LD estimate into an SFRD assuming a standard Salpeter initial mass function and including the unobscured contribution based on the UV dust-uncorrected emission from local galaxies, we estimate an SFRD scaling of SFRD0 + 0.08z, where SFRD0 ≃ (1.9 ± 0.03) × 10−2 [M⊙ Mpc−3] is our total SFRD estimate at z ∼ 0.02.