  We describe directed searches for continuous gravitational waves in data from
the sixth LIGO science data run. The targets were nine young supernova remnants
not associated with pulsars; eight of the remnants are associated with
non-pulsing suspected neutron stars. One target's parameters are uncertain
enough to warrant two searches, for a total of ten. Each search covered a broad
band of frequencies and first and second frequency derivatives for a fixed sky
direction. The searches coherently integrated data from the two LIGO
interferometers over time spans from 5.3-25.3 days using the matched-filtering
F-statistic. We found no credible gravitational-wave signals. We set 95%
confidence upper limits as strong (low) as $4\times10^{-25}$ on intrinsic
strain, $2\times10^{-7}$ on fiducial ellipticity, and $3\times10^{-6}$ on
r-mode amplitude. These beat the indirect limits from energy conservation and
are within the range of theoretical predictions for neutron-star ellipticities
and r-mode amplitudes.
