We investigate the high-frequency radio spectra of 20 low-luminosity active galactic nuclei (LLAGNs) with compact radio cores. Our millimetre survey with the Nobeyama Millimetre Array (NMA) and analyses of submillimetre archival data that had been obtained with the Submillimetre Common User Bolometer Array (SCUBA) on the James Clerk Maxwell Telescope (JCMT) reveal the following properties. At least half of the LLAGNs show inverted spectra between 15 and 96 GHz; we use the published data at 15 GHz with the Very Large Array (VLA) in a 0.15-arcsec resolution and our measurements at 96 GHz with the NMA in a 7-arcsec resolution. The inverted spectra are not artificially made due to their unmatched beam sizes, because of little diffuse contamination from dust, H ii regions, or extended jets in these LLAGNs. Such high-frequency inverted spectra are apparently consistent with a ‘submillimetre bump’, which is predicted by an advection-dominated accretion flow (ADAF) model. We find a strong correlation between the high-frequency spectral index and low-frequency core power measured with very-long-baseline-interferometry (VLBI) instruments. The inverted spectra are found exclusively in low-core-power sources, while steep spectra are in high-core-power ones with prominent pc-scale jets. This suggests that the ADAF and non-thermal jets may coexist. The flux ratios between disc and jet seem to be different from LLAGN to LLAGN; disc components can be seen in nuclear radio spectra only if the jets are faint.