We present the results of time-integrated searches for astrophysical neutrino sources in both the northern and southern skies. Data were collected using the partially completed IceCube detector in the 40-string configuration recorded between 2008 April 5 and 2009 May 20, totaling 375.5 days livetime. An unbinned maximum likelihood ratio method is used to search for astrophysical signals. The data sample contains 36,900 events: 14,121 from the northern sky, mostly muons induced by atmospheric neutrinos, and 22,779 from the southern sky, mostly high-energy atmospheric muons. The analysis includes searches for individual point sources and stacked searches for sources in a common class, sometimes including a spatial extent. While this analysis is sensitive to TeV-PeV energy neutrinos in the northern sky, it is primarily sensitive to neutrinos with energy greater than about 1 PeV in the southern sky. No evidence for a signal is found in any of the searches. Limits are set for neutrino fluxes from astrophysical sources over the entire sky and compared to predictions. The sensitivity is at least a factor of two better than previous searches (depending on declination), with 90% confidence level muon neutrino flux upper limits being between E(2)d Phi/dE similar to 2-200 x 10(-12) TeV cm(-2) s(-1) in the northern sky and between 3-700 x 10(-12) TeV cm(-2) s(-1) in the southern sky. The stacked source searches provide the best limits to specific source classes. The full IceCube detector is expected to improve the sensitivity to d Phi/dE proportional to E-2 sources by another factor of two in the first year of operation.