  The IceCube Neutrino Observatory is highly sensitive to neutrino bursts of
$\mathcal{O}$(10) MeV energy that are would be generated by core collapse
supernovae in our Galaxy. It will resolve temporal structures in supernova
light curves particularly well. In the light of an improved understanding of
the ice properties and the detector response, the effective area and the
corresponding uncertainties were newly determined with a Geant4-based Monte
Carlo. Uncertainties due to cross sections and oscillation effects in the Earth
were also investigated. This analysis has been extended by simulating a very
large sample to determine the small coincidence probability between optical
modules that bears information on the average neutrino energy. These simulation
results were then used to interpret the data in time and frequency space. While
the availability to record data for low energy neutrinos from supernovae is
close to perfect (99.2$\%$ between 2013-2020), the analysis requires that the
detector works faultlessly and artifacts do not mimic the signal in the 13
years of data taken so far. An effort has been made to keep the uptime after
all analysis steps similarly high. The frequency space can be studied in a
range between 1 Hz and 1/year to test the detector stability with high
accuracy, to study the influence of cosmic rays, and to search for periodic
phenomena that lead to sub-threshold increases in the count rates. Here we
discuss the results of the simulations and the corresponding systematic
limitations, the method to reconstruct the mean neutrino energy for a recorded
supernova, as well as aspects of the analyses of continuously taken optical
module rate data in the time and frequency domain.
