  We present the volumetric rates and luminosity functions (LFs) of Type Ia
supernovae (SNe Ia) from the $V$-band All-Sky Automated Survey for Supernovae
(ASAS-SN) catalogues spanning discovery dates from UTC 2014-01-26 to UTC
2017-12-29. Our standard sample consists of 404 SNe Ia with
$m_{V,\mathrm{peak}}<17$ mag and Galactic latitude $|b|>15^{\circ}$. Our
results are both statistically more precise and systematically more robust than
previous studies due to the large sample size and high spectroscopic
completeness. We make completeness corrections based on both the apparent and
absolute magnitudes by simulating the detection of SNe Ia in ASAS-SN light
curves. We find a total volumetric rate for all subtypes of
$R_{\mathrm{tot}}=2.28^{+0.20}_{-0.20}\,\times
10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$ for
$M_{V,\mathrm{peak}}<-16.5$ mag ($R_{\mathrm{tot}}=1.91^{+0.12}_{-0.12}\,\times
10^{4}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$ for
$M_{V,\mathrm{peak}}<-17.5$ mag) at the median redshift of our sample,
$z_{\mathrm{med}}=0.024$. This is in agreement ($1\sigma$) with the local
volumetric rates found by previous studies. We also compile luminosity
functions (LFs) for the entire sample as well as for subtypes of SNe Ia for the
first time. The major subtypes with more than one SN include Ia-91bg, Ia-91T,
Ia-CSM, and Ia-03fg with total rates of $R_{\mathrm{Ia-91bg}}=1.4^{+0.5}_{-0.5}
\times 10^{3}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$,
$R_{\mathrm{Ia-91T}}=8.5^{+1.6}_{-1.7} \times
10^{2}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$,
$R_{\mathrm{Ia-CSM}}=10^{+7}_{-7}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$,
and
$R_{\mathrm{Ia-03fg}}=30^{+20}_{-20}\,\mathrm{yr}^{-1}\,\mathrm{Gpc}^{-3}\,h^{3}_{70}$,
respectively. We estimate a mean host extinction of $E(V-r)\approx 0.2$ mag
based on the shift between our $V$-band and the ZTF $r$-band LFs.
