with flux density $S$ at frequency $\nu$, is used throughout this work). This suggests that a population of faint, flat-spectrum sources is emerging at flux densities $\lesssim 1 \textrm{ mJy}$ in the high-frequency sky.

In Paper I the spectral index distribution of this sample of sources selected at 15.7 GHz was compared to those of two samples selected at 1.4 GHz from FIRST and NVSS. This showed that there is a significant flat-spectrum population present in the 10C sample which is missing from the samples selected at 1.4 GHz. The 10C sample was compared to a sample of sources selected from the Square Kilometre Array Design Studies (SKADS) Simulated Sky (S$^3$) by @2008MNRAS.388.1335W [@2010MNRAS.405..447W] and we found that this simulation fails to reproduce the observed spectral index distribution and significantly under-predicts 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 Fanaroff and Riley type I (FRI; @1974MNRAS.167P..31F) sources becoming dominant at high frequencies. These results highlight that the faint, high-frequency source population is poorly understood and therefore the importance of further study of this population.

In this paper, we select all sources from 