The equation of state plays a critical role in the physics of the merger of two neutron stars. Recent numerical simulations with microphysical equation of state suggest the outcome of such events depends on the mass of the neutron stars. For less massive systems, simulations favor the formation of a hypermassive, quasi-stable neutron star, whose oscillations produce a short, high frequency burst of gravitational radiation. Its dominant frequency content is tightly correlated with the radius of the neutron star, and its measurement can be used to constrain the supranuclear equation of state. In contrast, the merger of higher mass systems results in prompt gravitational collapse to a black hole. We have developed an algorithm which combines waveform reconstruction from a morphology-independent search for gravitational wave transients with Bayesian model selection, to discriminate between post-merger scenarios and accurately measure the dominant oscillation frequency. We demonstrate the efficacy of the method using a catalogue of simulated binary merger signals in data from LIGO and Virgo, and we discuss the prospects for this analysis in advanced ground-based gravitational wave detectors. From the waveforms considered in this work and assuming an optimally oriented source, we find that the post-merger neutron star signal may be detectable by this technique to $sim 10text{--}25$,Mpc. We also find that we successfully discriminate between the post-merger scenarios with $sim 95%$ accuracy and determine the dominant oscillation frequency of surviving post-merger neutron stars to within $sim 10$,Hz, averaged over all detected signals. This leads to an uncertainty in the estimated radius of a non-rotating 1.6,M$_{odot}$ reference neutron star of $sim 100,$m.
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