  Gravitational waves from binary neutron star inspirals have been detected
along with the electromagnetic transients coming from the aftermath of the
merger in GW170817. However, much is still unknown about the post-merger
dynamics that connects these two sets of observables. This includes if, and
when, the post-merger remnant star collapses to a black hole, and what are the
necessary conditions to power a short gamma-ray burst, and other observed
electromagnetic counterparts. Observing the collapse of the post-merger neutron
star would shed led on these questions, constraining models for the short
gamma-ray burst engine and the hot neutron star equation of state. In this
work, we explore the scope of using gravitational wave detectors to measure the
timing of the collapse either indirectly, by establishing the shut-off of the
post-merger gravitational emission, or---more challengingly---directly, by
detecting the collapse signal. For the indirect approach, we consider a
kilohertz high-frequency detector design that utilises a previously studied
coupled arm cavity and signal recycling cavity resonance. This design would
give a signal-to-noise ratio of 0.5\,-\,8.6 (depending on the variation of
waveform parameters) for a collapse gravitational wave signal occurring at
10\,ms post-merger of a binary at 50\,Mpc and with total mass $2.7 M_\odot$.
For the direct approach, we propose a narrow-band detector design, utilising
the sensitivity around the frequency of the arm cavity free spectral range. The
proposed detector achieves a signal-to-noise ratio of 0.3\,-\,1.9, independent
of the collapse time. This detector is limited by both the fundamental
classical and quantum noise with the arm cavity power chosen as 10\,MW.
