The IceCube Collaboration has recently reported the observation of a flux of high-energy astrophysical neutrinos. The angular distribution of events is consistent with an isotropic arrival direction of neutrinos which is expected for an extragalactic origin. We estimate the prospects of detecting individual neutrino sources from a quasidiffuse superposition of many extragalactic sources at the level of the IceCube observation. Unlike previous analyses, we take into account ensemble variations of the source distribution as well as the event statistics of individual sources. We show that IceCube in its present configuration is sensitive to rare ≲10-8  Mpc-3 yr-1 transient source classes within five years of operation via the observation of event clusters. Identification of time-independent sources is more challenging due to larger backgrounds. We estimate that during the same period IceCube is sensitive to sparse sources with densities of ≲10-6  Mpc-3 via association of events with the closest 100 sources of an ensemble. We show that a next-generation neutrino observatory with 5 times the effective area of IceCube and otherwise similar detector performance would increase the sensitivity to source densities and rates by about 2 orders of magnitude.