What observations can be expected from the LIGO† gravitational wave observatory if and when Betelgeuse goes supernova?
Would we know that Betelgeuse has gone supernova before we see it light up our night sky?
†or any other current or soon-to-be completed gravitational wave obervatories.
Potentially, a short (less than a second) burst of gravitational waves (GWs) would be detected. Much depends on asymmetries in the core collapse, since a spherically symmetric collapse would not produce GWs (e.g. Morozova et al. 2019). However, theoretical models suggest that the GWs start at low frequency (tens of Hz) and are associated with the core bounce at the end of core collapse. After about half a second the frequencies rise to about 1kHz and are associated with oscillations of the proto-neutron star at the centre and then contunue to rise, but weaken in amplitude after that. Since LIGO is sensitive from about 20Hz to a few kHz, then only the initial phase that lasts a few tenths of a second would likely be detectable, but since GWs can travel through the star (and at the speed of light) without much hindrance they will indicate the moment of core collapse quite accurately.
The theoretical calculations referenced above suggest GW amplitudes of $h_{+}d \sim 10$, where $h_{+}$ is the dimensionless GW strain and $d$ is the distance to the supernova in cm. Betelgeuse is at a distance of $\sim 6 \times 10^{20}$ cm (600 light years), so the strain at the detector would be $h_{+} \sim 2\times 10^{-20}$. LIGO is sensitive to strains of above $10^{-23}$ in this frequency range, so there should be no problem (according to these models) in detecting such an event.
If they were produced with detectable strength, then they should be received at the same time as an anticipated neutrino burst. The difference would be that the neutrino burst should extend over about 10 s. That is because the dense centre of a supernova is opaque even to neutrinos (but not gravitational waves) for that period and so they "leak out" over that duration.
The electromagnetic signature of the supernova (i.e. a visible light increase) would be seen a few hours after that, since that is how long it takes for energy to be transferred to the envelope of the star.
NB: For all practical intents and purposes, over the distance to Betelgeuse, neutrinos, along with light and gravitational waves, travel at the same speed. i.e. The delay of the ($\sim 10$ MeV) neutrino burst due to their non-zero ($\ < 1$ eV) mass would be negligible. If the Lorentz factor is $10^7$, then the neutrino speed is the same as the speed of light to 1 part in $2\times 10^{14}$, which means they take an extra 0.1 ms to arrive from Betelgeuse compared to light speed.
Various issues in your question:
Betelgeuse is currently undergoing a "low-state" period, which is the combined effect of two periodic variations (See for instance http://www.astronomerstelegram.org/?read=13365). Hence, there is no clear evidence right now that it will blow up as a core-collapse supernova, say, wihtin the next months.
We are able to know if a core-collapse supernova occurs before optical light reaches us thanks to the detection of neutrinos. This was the case for the famous SN1987A in the Magellanic Clouds (https://en.wikipedia.org/wiki/SN_1987A#Neutrino_emissions, 2-3 hours before optical light)
LIGO is already operational (https://en.wikipedia.org/wiki/LIGO)
As far as I can tell, the merging of two neutron stars or two black holes releases more energy and produces a stronger gravitational wave that a "simple" collapse of a massive star (but this is a non-documented opinion). But Betelegeuse being a lot closer than the current detected sources of Gravitational Waves, this might be an interesting target indeed (although, see first point...)
