The IceCube Neutrino Observatory has been continuously taking data to search for O(0.5−10) s long neutrino bursts since 2007. Even if a Galactic core-collapse supernova is optically obscured or collapses to a black hole instead of exploding, it will be detectable via the O(10) MeV neutrino burst emitted during the collapse. We discuss a search for such events covering the time between April 17, 2008 and December 31, 2019. Considering the average data taking and analysis uptime of 91.7 % after all selection cuts, this is equivalent to 10.735 years of continuous data taking. In order to test the most conservative neutrino production scenario, the selection cuts were optimized for a model based on a 8.8 solar mass progenitor collapsing to an O-Ne-Mg core. Conservative assumptions on the effects of neutrino oscillations in the exploding star were made. The final selection cut was set to ensure that the probability to detect such a supernova within the Milky Way exceeds 99 %. No such neutrino burst was found in the data after performing a blind analysis. Hence, a 90 % C.L. upper limit on the rate of core-collapse supernovae out to distances of ≈ 25 kpc was determined to be 0.23/yr. For the more distant Magellanic Clouds, only high neutrino luminosity supernovae will be detectable by IceCube, unless external information on the burst time is available. We determined a model-independent limit by parameterizing the dependence on the neutrino luminosity and the energy spectrum.