  We study the shapes of galaxy dark matter haloes by measuring the anisotropy
of the weak gravitational lensing signal around galaxies in the second
Red-sequence Cluster Survey (RCS2). We determine the average shear anisotropy
within the virial radius for three lens samples: all galaxies with
19<m_r'<21.5, and the `red' and `blue' samples, whose lensing signals are
dominated by massive low-redshift early-type and late-type galaxies,
respectively. To study the environmental dependence of the lensing signal, we
separate each lens sample into an isolated and clustered part and analyse them
separately. We also measure the azimuthal dependence of the distribution of
physically associated galaxies around the lens samples. We find that these
satellites preferentially reside near the major axis of the lenses, and
constrain the angle between the major axis of the lens and the average location
of the satellites to <theta>=43.7 deg +/- 0.3 deg for the `all' lenses,
<theta>=41.7 deg +/- 0.5 deg for the `red' lenses and <theta>=42.0 deg +/- 1.4
deg for the `blue' lenses. For the `all' sample, we find that the anisotropy of
the galaxy-mass cross-correlation function <f-f_45>=0.23 +/- 0.12, providing
weak support for the view that the average galaxy is embedded in, and
preferentially aligned with, a triaxial dark matter halo. Assuming an
elliptical Navarro-Frenk-White (NFW) profile, we find that the ratio of the
dark matter halo ellipticity and the galaxy ellipticity
f_h=e_h/e_g=1.50+1.03-1.01, which for a mean lens ellipticity of 0.25
corresponds to a projected halo ellipticity of e_h=0.38+0.26-0.25 if the halo
and the lens are perfectly aligned. For isolated galaxies of the `all' sample,
the average shear anisotropy increases to <f-f_45>=0.51+0.26-0.25 and
f_h=4.73+2.17-2.05, whilst for clustered galaxies the signal is consistent with
zero. (abridged)
