  We report in detail on searches for eV-scale sterile neutrinos, in the
context of a 3+1 model, using eight years of data from the IceCube neutrino
telescope. By analyzing the reconstructed energies and zenith angles of 305,735
atmospheric $\nu_\mu$ and $\bar{\nu}_\mu$ events we construct confidence
intervals in two analysis spaces: $\sin^2 (2\theta_{24})$ vs. $\Delta m^2_{41}$
under the conservative assumption $\theta_{34}=0$; and $\sin^2(2\theta_{24})$
vs. $\sin^2 (2\theta_{34})$ given sufficiently large $\Delta m^2_{41}$ that
fast oscillation features are unresolvable. Detailed discussions of the event
selection, systematic uncertainties, and fitting procedures are presented. No
strong evidence for sterile neutrinos is found, and the best-fit likelihood is
consistent with the no sterile neutrino hypothesis with a p-value of 8\% in the
first analysis space and 19\% in the second.
