  The Neutrino Mass Ordering (NMO) remains one of the outstanding questions in
the field of neutrino physics. One strategy to measure the NMO is to observe
matter effects in the oscillation pattern of atmospheric neutrinos above $\sim
1\,\mathrm{GeV}$, as proposed for several next-generation neutrino experiments.
Moreover, the existing IceCube DeepCore detector can already explore this type
of measurement. We present rthe development and application of two independent
analyses to search for the signature of the NMO with three years of DeepCore
data. These analyses include a full treatment of systematic uncertainties and a
statistically-rigorous method to determine the significance for the NMO from a
fit to the data. Both analyses show that the dataset is fully compatible with
both mass orderings. For the more sensitive analysis, we observe a preference
for Normal Ordering with a $p$-value of $p_\mathrm{IO} = 15.3\%$ and
$\mathrm{CL}_\mathrm{s}=53.3\%$ for the Inverted Ordering hypothesis, while the
experimental results from both analyses are consistent within their
uncertainties. Since the result is independent of the value of
$\delta_\mathrm{CP}$ and obtained from energies $E_\nu \gtrsim
5\,\mathrm{GeV}$, it is complementary to recent results from long-baseline
experiments. These analyses set the groundwork for the future of this
measurement with more capable detectors, such as the IceCube Upgrade and the
proposed PINGU detector.
