  We report the results of a multimessenger search for coincident signals from
the LIGO and Virgo gravitational-wave observatories and the partially completed
IceCube high-energy neutrino detector, including periods of joint operation
between 2007-2010. These include parts of the 2005-2007 run and the 2009-2010
run for LIGO-Virgo, and IceCube's observation periods with 22, 59 and 79
strings. We find no significant coincident events, and use the search results
to derive upper limits on the rate of joint sources for a range of source
emission parameters. For the optimistic assumption of gravitational-wave
emission energy of $10^{-2}$\,M$_\odot$c$^2$ at $\sim 150$\,Hz with $\sim
60$\,ms duration, and high-energy neutrino emission of $10^{51}$\,erg
comparable to the isotropic gamma-ray energy of gamma-ray bursts, we limit the
source rate below $1.6 \times 10^{-2}$\,Mpc$^{-3}$yr$^{-1}$. We also examine
how combining information from gravitational waves and neutrinos will aid
discovery in the advanced gravitational-wave detector era.
