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Neutrinos are the ideal cosmic messengers, and can be used to explore the most powerful accelerators in the Universe, in particular the mechanisms for producing and accelerating cosmic rays to incredible energies. By studying clustering of neutrino candidate events in the IceCube detector we can discover sites of hadronic acceleration. We present results on searches for point-like sources of astrophysical neutrinos located in the Southern hemisphere, at energies between 100 GeV and a few TeV. The data were collected during the first year of the completed 86-string detector, corresponding to a detector livetime of 329 days. The event selection focuses on identifying events starting inside the instrumented volume, utilizing several advanced veto techniques, successfully reducing the large background of atmospheric muons. An unbinned maximum likelihood method is used to search for clustering of neutrino-like events. We perform a search in the full Southern hemisphere and a dedicated search using a catalog of 96 pre-defined known gamma-ray emitting sources seen in ground-based telescopes. No evidence of neutrino emission from point-like sources is found. The hottest spot is located at R.A. 305.2° and Dec. -8.5°, with a post-trial p-value of 88.1%. The most significant source in the a priori list is QSO 2022-077 with a post-trial p-value of 14.8%. In the absence of evidence for a signal, we calculate upper limits on the flux of muon-neutrinos for a range of spectra. For an unbroken E^-2 neutrino spectrum, the observed limits are between 2.8 and 9.4×10^-10 TeV cm^-2 s^-1, while for an E^-2 neutrino spectrum with an exponential cut-off at 10 TeV, the observed limits are between 0.6 and 3.6×10^-9 TeV cm^-2 s^-1.