We have studied a sample of 296 faint (>0.5mJy) radio sources selected from an area of the Tenth Cambridge (10C) survey at 15.7GHz in the Lockman Hole. By matching this catalogue to several lower frequency surveys (e.g. including a deep GMRT survey at 610MHz, a WSRT survey at 1.4GHz, NVSS, FIRST and WENSS) we have investigated the radio spectral properties of the sources in this sample; all but 30 of the 10C sources are matched to one or more of these surveys. We have found a significant increase in the proportion of flat spectrum sources at flux densities below 1mJy - the median spectral index between 15.7GHz and 610MHz changes from 0.75 for flux densities greater than 1.5mJy to 0.08 for flux densities less than 0.8mJy. This suggests that a population of faint, flat spectrum sources is emerging at flux densities around 1mJy. The spectral index distribution of this sample of sources selected at 15.7GHz is compared to those of two samples selected at 1.4GHz from FIRST and NVSS. We find that there is a significant flat spectrum population present in the 10C sample which is missing from the samples selected at 1.4GHz. The 10C sample is compared to a sample of sources selected from the SKADS Simulated Sky by Wilman et al. (2008MNRAS.388.1335W) and we find that this simulation fails to reproduce the observed spectral index distribution and significantly underpredicts the number of sources in the faintest flux density bin. It is likely that the observed faint, flat spectrum sources are a result of the cores of FRI sources becoming dominant at high frequencies. These results highlight the importance of studying this faint, high frequency population.