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Core-collapse supernovae are remarkable astrophysical phenomena emitting optical photons, gamma rays, neutrinos and in some scenarios, gravitational waves. Due to their weakly interacting nature, neutrinos can escape the dense core of a collapsing star, where electromagnetic radiation is trapped. Therefore, they carry direct information about the hydrodynamics and energy transport processes that govern the supernova explosion. A high-statistics neutrino observation is required to uncover the explosion mechanism and to constrain key physical parameters of the collapsing core. In particular, processes related to the shock-revival can produce rapid modulations of the neutrino luminosity and energy. 

The IceCube Neutrino Observatory is a cubic kilometre telescope located at the South Pole. Due to its size, IceCube is highly sensitive to the low-energy, MeV neutrino burst from a supernova. When these neutrinos interact with the ice, they generate Cherenkov light, which is measured by an array of 5,160 optical modules. However, due to the high background noise from the optical sensors and the wide spacing between modules, individual neutrinos cannot be reconstructed. Instead, IceCube searches for a collective excess of the detection rate above the noise floor. In the long term, the IceCube will evolve into the next-generation neutrino telescope, called IceCube-Gen2. Encompassing, among other things, an optical array nearly eight times the size of the current IceCube and around 10,000 new sensors, IceCube-Gen2 will have unmatched statistics for nearby supernovae.

In this thesis, we present results on the projected sensitivity of IceCube and IceCube-Gen2 in observing features in the neutrino light curve. By discussing the potential enhancement of wavelength shifters, we inform the decision-making process for future detector design. Furthermore, we have developed a framework for detailed simulations of supernova neutrino interactions and radioactive decays. This is the first step towards a full neutrino energy reconstruction using machine learning tools.