This paper presents four searches for flaring sources of neutrinos using the IceCube neutrino telescope. For the first time, a search is performed over the entire parameter space of energy, direction, and time with sensitivity to neutrino flares lasting between 20 mu s and a year duration from astrophysical sources. Searches that integrate over time are less sensitive to flares because they are affected by a larger background of atmospheric neutrinos and muons that can be reduced by the use of additional timing information. Flaring sources considered here, such as active galactic nuclei, soft gamma-ray repeaters, and gamma-ray bursts, are promising candidate neutrino emitters. Two searches are "untriggered" in the sense that they look for any possible flare in the entire sky and from a predefined catalog of sources from which photon flares have been recorded. The other two searches are triggered by multi-wavelength information on flares from blazars and from a soft gamma-ray repeater. One triggered search uses lightcurves from Fermi-LAT which provides continuous monitoring. A second triggered search uses information where the flux states have been measured only for short periods of time near the flares. The untriggered searches use data taken by 40 strings of IceCube between 2008 April 5 and 2009 May 20. The triggered searches also use data taken by the 22-string configuration of IceCube operating between 2007 May 31 and 2008 April 5. The results from all four searches are compatible with a fluctuation of the background.