  We study the gravitational lensing effects of spiral galaxies by taking a
model of the Milky Way and computing its lensing properties. The model is
composed of a spherical Hernquist bulge, a Miyamoto-Nagai disc and an
isothermal halo. As a strong lens, a spiral galaxy like the Milky Way can give
rise to four different imaging geometries. They are (i) three images on one
side of the galaxy centre (`disc triplets'), (ii) three images with one close
to the centre (`core triplets'), (iii) five images and (iv) seven images.
Neglecting magnification bias, we show that the core triplets, disc triplets
and fivefold imaging are roughly equally likely. Even though our models contain
edge-on discs, their image multiplicities are not dominated by disc triplets.
The halo has a small effect on the caustic structure, the time delays and
brightnesses of the images. The Milky Way model has a maximum disc (i.e., the
halo is not dynamically important in the inner parts). Strong lensing by nearly
edge-on disc galaxies breaks the degeneracy between the relative contribution
of the disc and halo to the overall rotation curve. If a spiral galaxy has a
sub-maximum disc, then the astroid caustic shrinks dramatically in size, whilst
the radial caustic shrinks more modestly. This causes changes in the relative
likelihood of the image geometries, specifically (i) core triplets are now 9/2
times more likely than disc triplets, (ii) the cross section for threefold
imaging is reduced by a factor of 2/3, whilst (iii) the cross section for
fivefold imaging is reduced by 1/2. Although multiple imaging is less likely
(the cross sections are smaller), the average total magnification is greater.
