  When studying transiting exoplanets it is common to assume a spherical planet
shape. However short rotational periods can cause a planet to bulge at its
equator, as is the case with Saturn whose equatorial radius is almost 10%
larger than its polar radius. As a new generation of instruments comes online,
it is important to continually assess the underlying assumptions of models to
ensure robust and accurate inferences. We analyze bulk samples of known
transiting planets and calculate their expected signal strength if they were to
be oblate. We find that for noise levels below 100ppm, as many as 100 planets
could have detectable oblateness. We also investigate the effects of fitting
spherical planet models to synthetic oblate lightcurves. We find that this
biases the retrieved parameters by several standard deviations for oblateness
values > 0.1-0.2. When attempting to fit an oblateness model to both spherical
and oblate lightcurves, we find that the sensitivity of such fits is correlated
with both the SNR as well as the time sampling of the data, which can mask the
oblateness signal. For typical values of these quantities for Kepler
observations, it is difficult to rule out oblateness values less than ~0.25.
This results in an accuracy wall of 10-15% for the density of planets which may
be oblate. Finally, we find that a precessing oblate planet has the ability to
mimic the signature of a long-period companion via transit timing variations,
inducing offsets at the level of 10s of seconds.
