  Longstanding anomalies in neutrino oscillation experiments point to the
existence of a fourth, hypothetical neutrino: the sterile neutrino. Global fits
to a sterile neutrino model find a strong preference for such a model over the
massive neutrino Standard Model. However, the fit results suffer from
inconsistencies between datasets, referred to as tension. This motivates more
complicated models for new physics. This thesis considers a model of unstable
sterile neutrinos, where the heaviest mass state can decay. First, the
phenomenology of unstable sterile neutrinos is explored in the IceCube
experiment, a gigaton neutrino detector located at the South Pole. Second,
global fits to traditional and unstable sterile neutrino models are combined
with one year of data from IceCube. A preference for the unstable sterile
neutrino model is found, as well as a reduction in tension. Lastly, a high
statistics search for unstable sterile neutrinos is performed in IceCube. The
Standard Model is rejected with a $p$-value of 2.8% and the traditional sterile
neutrino model is rejected with a $p$-value of 4.9%. The best-fit point is
$\Delta m_{41}^2 = 6.7^{+3.9}_{-2.5}$ eV$^{2}$, $\sin^2 2 \theta_{24} =
0.33^{+0.20}_{-0.17}$, and $g^2 = 2.5 \pi \pm 1.5 \pi$, where $g$ is the
coupling that mediates the neutrino decay. The best-fit corresponds to a
lifetime of the heaviest neutrino of $\tau_4/m_4 = 6\times 10^{-16}$ s/eV. A
Bayesian analysis finds a best model with similar sterile parameters.
