  Recent studies of the two-dimensional, classical $XY$ magnet in a magnetic
field suggest that it has three distinct vortex phases: a linearly confined
phase, a logarithmically confined phase, and a free vortex phase. In this work
we study spin-spin correlation functions in this model by analytical analysis
and numerical simulations to search for signatures of the various phases. In
all three phases, the order parameter is nonzero and $<\cos(\th({\bf
r}_1))\cos(\th({\bf r}_2))>$ remains nonzero for $r \equiv |{\bf r}_1-{\bf
r}_2|\rightarrow \infty$, indicating the expected long range order. The
correlation function for transverse fluctuations of the spins,
$C(r)=<\sin(\th({\bf r}_1))\sin(\th({\bf r}_2))>$, falls exponentially in all
three phases. A renormalization group analysis suggests that the
logarithmically confined phase should have a spatially anisotropic correlation
length. In addition, there is a generic anisotropy in the prefactor which is
always present. We find that this prefactor anisotropy becomes rather strong in
the presence of a magnetic field, masking the effects of any anisotropy in the
correlation length in the simulations.
