  The density correlations of some singular Fermi liquids with anomalous
properties such as resistivity varying linearly with T at low temperatures, a
$T \log T$ contribution to the entropy and thermopower, etc., are expected to
be quite different from that in Landau Fermi liquids. A possible statistical
mechanical model for the quantum critical fluctuations in diverse systems with
such properties is the 2D dissipative quantum XY model. Exact relations between
the density correlations and singular irreducible vertices due to coupling of
Fermions to the topological excitations of the 2D dissipative quantum XY model
are used to derive results which were proposed phenomenologically long ago but
are measurable only recently due to advances in experimental techniques. The
density correlations are unusual at all momenta ${\bf q}$ and energy $\nu$,
from the hydrodynamic limit to that for large momenta and energy. The
hydrodynamic limit together with the continuity equation gives the linear in T
resistivity. The density correlations are almost independent of frequency up to
a high frequency cut-off for $q_{ZB} \gtrsim q >> \nu/v_F$; $q_{ZB}$ is the
Brillouin zone boundary and $v_F$ is the Fermi-velocity. The results should be
applicable to loop-current quantum criticality in cuprates, and to 2D Fe based
compounds near their antiferromagnetic quantum-criticality. The relation of the
results to the temperature and frequency dependent conductivity and to Raman
response is also discussed.
