  We present the results of a systematic search for gravitationally-lensed arcs
in clusters of galaxies located in the Hubble Space Telescope Wide Field and
Planetary Camera 2 data archive. By carefully examining the images of 128
clusters we have located 12 candidate radial arcs and 104 tangential arcs, each
of whose length to width ratio exceeds 7. In addition, 24 other radial arc
candidates were identified with a length to width ratio less than 7. Keck
spectroscopy of 17 candidate radial arcs suggests that contamination of the
radial arc sample from non-lensed objects is ~30-50%. With our catalog, we
explore the practicality of using the number ratio of radial to tangential arcs
as a statistical measure of the slope \beta of the dark matter distribution in
cluster cores (where \rho_{DM}\propto r^{-\beta} at small radii). Despite the
heterogeneous nature of the cluster sample, we demonstrate that this abundance
ratio is fairly constant across various cluster subsamples partitioned
according to X-ray luminosity and optical survey depth. We develop the
necessary formalism to interpret this ratio in the context of two-component
mass models for cluster cores. Although the arc statistics in our survey are
consistent with a range of density profiles -- \beta<~1.6 depending on various
assumptions, we show that one of the prime limiting factors is the distribution
of stellar masses for the brightest cluster galaxies. We discuss the prospects
for improving the observational constraints and thereby providing a reliable
statistical constraint on cluster dark matter profiles on <~100 kpc scales.
