A search for high-energy neutrinos was performed using data collected by the IceCube Neutrino Observatory from May 2009 to May 2010, when the array was running in its 59-string configuration. The data sample was optimized to contain muon neutrino induced events with a background contamination of atmospheric muons of less than 1%. These data, which are dominated by atmospheric neutrinos, are analyzed with a global likelihood fit to search for possible contributions of prompt atmospheric and astrophysical neutrinos, neither of which have yet been identified. Such signals are expected to follow a harder energy spectrum than conventional atmospheric neutrinos. In addition, the zenith angle distribution differs for astrophysical and atmospheric signals. A global fit of the reconstructed energies and directions of observed events is performed, including possible neutrino flux contributions for an astrophysical signal and atmospheric backgrounds as well as systematic uncertainties of the experiment and theoretical predictions. The best fit yields an astrophysical signal flux for ${\ensuremath{\nu}}_{\ensuremath{\mu}}+{\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$ of ${E}^{2}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{\ensuremath{\Phi}}(E)=0.25\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}\text{ }\text{ }\mathrm{GeV}\text{ }{\mathrm{cm}}^{\ensuremath{-}2}\text{ }{\mathrm{s}}^{\ensuremath{-}1}\text{ }{\mathrm{sr}}^{\ensuremath{-}1}$, and a zero prompt component. Although the sensitivity of this analysis for astrophysical neutrinos surpasses the Waxman and Bahcall upper bound, the experimental limit at 90% confidence level is a factor of 1.5 above at a flux of ${E}^{2}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{\ensuremath{\Phi}}(E)=1.44\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}\text{ }\text{ }\mathrm{GeV}\text{ }{\mathrm{cm}}^{\ensuremath{-}2}\text{ }{\mathrm{s}}^{\ensuremath{-}1}\text{ }{\mathrm{sr}}^{\ensuremath{-}1}$.