Context. Deep mid-infrared (MIR) surveys have revealed numerous strongly star-forming galaxies at redshift z ≲ 2. Their MIR fluxes are produced by a combination of continuum and polycyclic aromatic hydrocarbon (PAH) emission features. The PAH features can dominate the total MIR flux, but are difficult to measure without spectroscopy. Aims. We aim to study star-forming galaxies by using a blind spectroscopic survey at MIR wavelengths to understand evolution of their star formation rate (SFR) and specific SFR (SFR per stellar mass) up to z ≃ 0.5, by paying particular attention to their PAH properties. Methods. We conducted a low-resolution (R ≃ 50) slitless spectroscopic survey at 5–13 μm of 9 μm flux-selected sources (>0.3 mJy) around the north ecliptic pole with the infrared camera (IRC) onboard AKARI. After removing 11 AGN candidates by using the IRC photometry, we identify 48 PAH galaxies with PAH 6.2, 7.7, and 8.6 μm features at z < 0.5. The rest-frame optical–MIR spectral energy distributions (SEDs) based on CFHT and IRC imaging covering 0.37–18 μm were produced, and analysed in conjunction with the PAH spectroscopy. We defined the PAH enhancement by using the luminosity ratio of the 7.7 μm PAH feature over the 3.5 μm stellar component of the SEDs. Results. The rest-frame SEDs of all PAH galaxies have a universal shape with stellar and 7.7 μm bumps, except that the PAH enhancement significantly varies as a function of the PAH luminosities. We identify a PAH-enhanced population at z ≳ 0.35, whose SEDs and luminosities are typical of luminous infrared galaxies. They show particularly larger PAH enhancement at high luminosity, implying that they are vigorous star-forming galaxies with elevated specific SFR. Our composite starburst model that combines a very young and optically very thick starburst with a very old population can successfully reproduce most of their SED characteristics, although we cannot confirm this optically think component from our spectral analysis.