  Models are developed to simulate lightcurves of stars dimmed by transiting
exoplanets with and without rings. These models are then applied to
\textit{Kepler} photometry to search for planetary rings in a sample of 21
exoplanets, mostly hot Jupiters, chosen to offer the best observational
opportunity for discovering potential rings. We also examine what kinds of
rings might be expected for these planets, in terms of both size and
orientation, based on arguments involving the host planet's equilibrium
temperature, its likely obliquities, and the formation and stability of
possible ring systems. Finding no evidence for rings, for each of the 21
studied planets it is determined on an observational basis which potential
rings can be rejected out of a representative set of fiducial rings, varying in
both size and orientation. For 12 of the 21 planets, we determined that
Saturn-like rings could be ruled out for at least certain orientations.
Additionally, the detectability of rings is studied, and it is found that
ringed planets with small obliquities (roughly $5^{\circ}-10^{\circ}$) can
yield large signals, which is encouraging for future work, since such small
obliquities are expected for hot Jupiters.
