  The possible formation histories of neutron star binaries remain unresolved
by current gravitational-wave catalogs. The detection of an eccentric binary
system could be vital in constraining compact binary formation models. We
present the first search for aligned spin eccentric neutron star-black hole
binaries (NSBH) and the most sensitive search for aligned-spin eccentric binary
neutron star (BNS) systems using data from the third observing run of the
advanced LIGO and advanced Virgo detectors. No new statistically significant
candidates are found; we constrain the local merger rate to be less than 150
$\text{Gpc}^{-3}\text{Yr}^{-1}$ for binary neutron stars in the field, and, 50,
100, and 70 $\text{Gpc}^{-3}\text{Yr}^{-1}$ for neutron star-black hole
binaries in globular clusters, hierarchical triples and nuclear clusters,
respectively, at the 90$\%$ confidence level if we assume that no sources have
been observed from these populations. We predict the capabilities of upcoming
and next-generation observatory networks; we investigate the ability of three
LIGO ($\text{A}^{#}$) detectors and Cosmic Explorer CE (20km) + CE (40km) to
use eccentric binary observations for determining the formation history of
neutron star binaries. We find that 2 -- 100 years of observation with three
$\text{A}^{#}$ observatories are required before we observe clearly eccentric
NSBH binaries; this reduces to only 10 days -- 1 year with the CE detector
network. CE will observe tens to hundreds of measurably eccentric binaries from
each of the formation models we consider.
