  We consider limits on the local ($z=0$) density ($n_0$) of extragalactic
neutrino sources set by the nondetection of steady high-energy neutrino sources
producing $\gtrsim50$ TeV muon multiplets in the present IceCube data, taking
into account the redshift evolution, luminosity function and neutrino spectrum
of the sources. We show that the lower limit depends moderately on source
spectra and strongly on redshift evolution. We find
$n_0\gtrsim{10}^{-8}-{10}^{-7}~{\rm Mpc}^{-3}$ for standard candle sources
evolving rapidly, $n_s\propto{(1+z)}^3$, and
$n_0\gtrsim{10}^{-6}-{10}^{-5}~{\rm Mpc}^{-3}$ for nonevolving sources. The
corresponding upper limits on their neutrino luminosity are $L_{{\nu_\mu}}^{\rm
eff}\lesssim10^{42}-10^{43}~{\rm erg}~{\rm s}^{-1}$ and $L_{{\nu_\mu}}^{\rm
eff}\lesssim10^{41}-10^{42}~{\rm erg}~{\rm s}^{-1}$, respectively. Applying
these results to a wide range of classes of potential sources, we show that
powerful blazar jets associated with active galactic nuclei are unlikely to be
the dominant sources. For almost all other steady candidate source classes
(including starbursts, radio galaxies, and galaxy clusters and groups), an
order of magnitude increase in the detector sensitivity at $\sim0.1-1$ PeV will
enable a detection (as point sources) of the few brightest objects. Such an
increase, which may be provided by next-generation detectors like IceCube-Gen2
and an upgraded KM3NET, can improve the limit on $n_0$ by more than two orders
of magnitude. Future gamma-ray observations (by Fermi, HAWC and CTA) will play
a key role in confirming the association of the neutrinos with their sources.
