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Many short gamma-ray bursts (GRBs) originate from binary neutron star mergers, and there are several theories that predict the production of coherent, prompt radio signals either prior, during, or shortly following the merger, as well as persistent pulsar-like emission from the spin-down of a magnetar remnant. Here we present a low frequency (170–200 MHz) search for coherent radio emission associated with nine short GRBs detected by the Swift and/or Fermi satellites using the Murchison Widefield Array (MWA) rapid-response observing mode. The MWA began observing these events within 30–60 s of their high-energy detection, enabling us to capture any dispersion delayed signals emitted by short GRBs for a typical range of redshifts. We conducted transient searches at the GRB positions on timescales of 5 s, 30 s, and 2 min, resulting in the most constraining flux density limits on any associated transient of 0.42, 0.29, and 0.084 Jy, respectively. We also searched for dispersed signals at a temporal and spectral resolution of 0.5 s and 1.28 MHz, but none were detected. However, the fluence limit of 80–100 Jy ms derived for GRB 190627A is the most stringent to date for a short GRB. Assuming the formation of a stable magnetar for this GRB, we compared the fluence and persistent emission limits to short GRB coherent emission models, placing constraints on key parameters including the radio emission efficiency of the nearly merged neutron stars (
$\epsilon_r\lesssim10^{-4}$
), the fraction of magnetic energy in the GRB jet (
$\epsilon_B\lesssim2\times10^{-4}$
), and the radio emission efficiency of the magnetar remnant (
$\epsilon_r\lesssim10^{-3}$
). Comparing the limits derived for our full GRB sample (along with those in the literature) to the same emission models, we demonstrate that our fluence limits only place weak constraints on the prompt emission predicted from the interaction between the relativistic GRB jet and the interstellar medium for a subset of magnetar parameters. However, the 30-min flux density limits were sensitive enough to theoretically detect the persistent radio emission from magnetar remnants up to a redshift of 
$z\sim0.6$
. Our non-detection of this emission could imply that some GRBs in the sample were not genuinely short or did not result from a binary neutron star merger, the GRBs were at high redshifts, these mergers formed atypical magnetars, the radiation beams of the magnetar remnants were pointing away from Earth, or the majority did not form magnetars but rather collapse directly into black holes.
The GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) is a radio continuum survey at 76–227 MHz of the entire southern sky (Declination 
$<\!{+}30^{\circ}$
) with an angular resolution of 
${\approx}2$
arcmin. In this paper, we combine GLEAM data with optical spectroscopy from the 6dF Galaxy Survey to construct a sample of 1 590 local (median 
$z \approx 0.064$
) radio sources with 
$S_{200\,\mathrm{MHz}} > 55$
mJy across an area of 
${\approx}16\,700\,\mathrm{deg}^{2}$
. From the optical spectra, we identify the dominant physical process responsible for the radio emission from each galaxy: 73% are fuelled by an active galactic nucleus (AGN) and 27% by star formation. We present the local radio luminosity function for AGN and star-forming (SF) galaxies at 200 MHz and characterise the typical radio spectra of these two populations between 76 MHz and 
${\sim}1$
GHz. For the AGN, the median spectral index between 200 MHz and 
${\sim}1$
GHz, 
$\alpha_{\mathrm{high}}$
, is 
$-0.600 \pm 0.010$
(where 
$S \propto \nu^{\alpha}$
) and the median spectral index within the GLEAM band, 
$\alpha_{\mathrm{low}}$
, is 
$-0.704 \pm 0.011$
. For the SF galaxies, the median value of 
$\alpha_{\mathrm{high}}$
is 
$-0.650 \pm 0.010$
and the median value of 
$\alpha_{\mathrm{low}}$
is 
$-0.596 \pm 0.015$
. Among the AGN population, flat-spectrum sources are more common at lower radio luminosity, suggesting the existence of a significant population of weak radio AGN that remain core-dominated even at low frequencies. However, around 4% of local radio AGN have ultra-steep radio spectra at low frequencies (
$\alpha_{\mathrm{low}} < -1.2$
). These ultra-steep-spectrum sources span a wide range in radio luminosity, and further work is needed to clarify their nature.
