  We present a catalogue of local ($z<0.1$) galaxies that contain faint AGN. We
select these objects by identifying galaxies that exhibit a significant excess
in their radio luminosities, compared to what is expected from the observed
levels of star-formation activity in these systems. This is achieved by
comparing the optical (spectroscopic) star formation rate (SFR) to the 1.4~GHz
luminosity measured from the FIRST survey. The majority of the AGN identified
in this study are fainter than those in previous work, such as in the Best and
Heckman (2012) catalogue. We show that these faint AGN make a non-negligible
contribution to the radio luminosity function at low luminosities (below
$10^{22.5}$ W Hz$^{-1}$), and host $\sim$13 per cent of the local radio
luminosity budget. Their host galaxies are predominantly high stellar-mass
systems (with a median stellar mass of $10^{11}\textrm{M}_{\odot}$), are found
across a range of environments (but typically in denser environments than
star-forming galaxies) and have early-type morphologies. This study
demonstrates a general technique to identify AGN in galaxy populations where
reliable optical SFRs can be extracted using spectro-photometry and where radio
data are also available so that a radio excess can be measured. Our results
also demonstrate that it is unsafe to infer SFRs from radio emission
\textit{alone}, even if bright AGN have been excluded from a sample, since
there is a significant population of faint radio AGN which may contaminate the
radio-derived SFRs.
