The IceCube Neutrino Observatory is a large Cherenkov detector integrated into 1km3 of Antarctic ice. Besides the detection of highly energetic astrophysical neutrinos, the detector can be used to search for signatures of exotic physics. This work is the search for subrelativistic, magnetic monopoles as remnants of the GUT era (Grand Unified Theory) shortly after the Big Bang. These monopoles can be detected by the Cherenkov light from nucleon decays, which are catalyzed via the Rubakov-Callan effect along the trajectory of the monopole. In this paper the results of the analysis of first data taken from May 2011 until May 2012 with a dedicated slow-particle trigger for DeepCore, a subdetector of IceCube, are presented. For the brightest monopoles a different analysis technique is applied. We present the results of a corresponding analysis applied to data taken from May 2009 until May 2010 with the 59-string configuration. In both analyses no monopole signal has been observed and their flux can be constrained to a level three orders of magnitude below the Parker bound. These results improve the current best experimental limits by more than one order of magnitude for a wide parameter space of monopole velocity and catalysis cross section. Since May 2012 a dedicated trigger is running on the full IceCube detector and multiple improvements for future searches for subrelativistic magnetic monopoles have been developed. Based on this the sensitivities of a future analysis are presented.