  What determines the nuclear radio emission in local galaxies? We combine
optical [O III] line emission, robust black hole (BH) mass estimates, and
high-resolution e-MERLIN 1.5-GHz data, from the LeMMINGs survey, of a
statistically-complete sample of 280 nearby, optically active (LINER and
Seyfert) and inactive HII and Absorption line galaxies [ALG]) galaxies. Using
[O III] luminosity ($L_{\rm [O~III]}$) as a proxy for the accretion power,
local galaxies follow distinct sequences in the optical-radio planes of BH
activity, which suggest different origins of the nuclear radio emission for the
optical classes. The 1.5-GHz radio luminosity of their parsec-scale cores
($L_{\rm core}$) is found to scale with BH mass ($M_{\rm BH}$) and [O~III]
luminosity. Below $M_{\rm BH} \sim$10$^{6.5}$ M$_{\odot}$, stellar processes
from non-jetted HII galaxies dominate with $L_{\rm core} \propto M_{\rm
BH}^{0.61\pm0.33}$ and $L_{\rm core} \propto L_{\rm [O~III]}^{0.79\pm0.30}$.
Above $M_{\rm BH} \sim$10$^{6.5}$ M$_{\odot}$, accretion-driven processes
dominate with $L_{\rm core} \propto M_{\rm BH}^{1.5-1.65}$ and $L_{\rm core}
\propto L_{\rm [O~III]}^{0.99-1.31}$ for active galaxies: radio-quiet/loud
LINERs, Seyferts and jetted HII galaxies always display (although low)
signatures of radio-emitting BH activity, with $L_{\rm 1.5\,
GHz}\gtrsim$10$^{19.8}$ W Hz$^{-1}$ and $M_{\rm BH}\gtrsim10^{7}$ M$_{\odot}$,
on a broad range of Eddington-scaled accretion rates ($\dot{m}$). Radio-quiet
and radio-loud LINERs are powered by low-$\dot{m}$ discs launching
sub-relativistic and relativistic jets, respectively. Low-power slow jets and
disc/corona winds from moderately high to high-$\dot{m}$ discs account for the
compact and edge-brightened jets of Seyferts, respectively. Jetted HII galaxies
may host weakly active BHs. Fuel-starved BHs and recurrent activity account for
ALG properties. [abridged]
