  Detection of gravitational waves from accreting neutron stars (NSs) in our
galaxy, due to ellipticity or internal oscillation, would be a breakthrough in
our understanding of compact objects and explain the absence of NSs rotating
near the break-up limit. Direct detection, however, poses a formidable
challenge. Using the current data available on the properties of the accreting
NSs in Low Mass X-Ray Binaries (LMXBs), we quantify the detectability for the
known accreting NSs, considering various emission scenarios and taking into
account the negative impact of parameter uncertainty on the data analysis
process. Only a few of the persistently bright NSs accreting at rates near the
Eddington limit are detectable by Advanced LIGO if they are emitting
gravitational waves at a rate matching the torque from accretion. A larger
fraction of the known population is detectable if the spin and orbital
parameters are known in advance, especially with the narrow-band Advanced LIGO.
We identify the most promising targets, and list specific actions that would
lead to significant improvements in detection probability. These include
astronomical observations (especially for unknown orbital periods),
improvements in data analysis algorithms and capabilities, and further detector
development.
