In this paper searches for flaring astrophysical neutrino sources and sources with periodic emission with the IceCube neutrino telescope are presented. In contrast to time-integrated searches, where steady emission is assumed, the analyses presented here look for a time-dependent signal of neutrinos using the information from the neutrino arrival times to enhance the discovery potential. A search was performed for correlations between neutrino arrival times and directions, as well as neutrino emission following time-dependent light curves, sporadic emission, or periodicities of candidate sources. These include active galactic nuclei, soft γ-ray repeaters, supernova remnants hosting pulsars, microquasars, and X-ray binaries. The work presented here updates and extends previously published results to a longer period that covers 4 years of data from 2008 April 5 to 2012 May 16, including the first year of operation of the completed 86 string detector. The analyses did not find any significant time-dependent point sources of neutrinos, and the results were used to set upper limits on the neutrino flux from source candidates.