Despite more than a hundred years of observations, the cosmic-ray puzzle is still one of the most intriguing open questions in astrophysics. In the last decade, a multi-messenger approach has become available to address this problem: in this regard, neutrinos are ideal messengers, propagating undeflected by magnetic fields and unimpeded by particle interactions. This thesis presents searches for transient astrophysical neutrino sources using 10 years of data collected by the IceCube experiment from April 2008 to July 2018. These data are optimised for muon track-like events with TeV–PeV energies, that provide the best angular resolution. The searches consist of hypothesis tests based on an unbinned maximum-likelihood ratio. A multi-flare algorithm has been developed to reconstruct multiple flares from the same direction in the sky. This method is used to test multi-flare neutrino emission from a catalogue of gamma-ray emitters, mainly comprising active galactic nuclei, and from the whole sky. For both searches, a population study is additionally performed to look for an astrophysical signal from an ensemble of subthreshold sources. Multi-flare neutrino emission is tested from the direction of the blazar J0242+1101, as part of a follow-up program triggered by recent results presented by the ANTARES collaboration. A single-flare analysis is performed to search for neutrino emission from the prompt and targeted X-ray afterglow phases of gamma-ray bursts. For this analysis, a stacking method is also performed and improved with respect to previous analyses. These improvements consist of the introduction of physically motivated weights, and of the use of consolidated empirical correlations of the gamma-ray burst parameters during the prompt phase to infer the luminosity and Lorentz factor, when redshift measurements are not available. No significant neutrino emission has been observed from the aforementioned searches. Evidence at the 3-sigma level is provided from the multi-flare population study of the catalogue. M87 emerges as the most significant transient source, while TXS 0506+056 is the only multi-flare source of the catalogue, with the two prominent flares reconstructed for the first time by the same analysis. The non-observation of a significant neutrino emission is used to set upper limits on the neutrino fluence in each search. Furthermore, relevant parameters of the single-zone fireball model of gamma-ray bursts are constrained.