  We prove variation-norm estimates for certain oscillatory integrals related
to Carleson's theorem. Bounds for the corresponding maximal operators were
first proven by Stein and Wainger. Our estimates are sharp in the range of
exponents, up to endpoints. Such variation-norm estimates have applications to
discrete analogues and ergodic theory. The proof relies on square function
estimates for Schr\"odinger-like equations due to Lee, Rogers and Seeger. In
dimension one, our proof additionally relies on a local smoothing estimate.
Though the known endpoint local smoothing estimate by Rogers and Seeger is more
than sufficient for our purpose, we also give a proof of certain local
smoothing estimates using Bourgain-Guth iteration and the Bourgain-Demeter
$\ell^2$ decoupling theorem. This may be of independent interest, because it
improves the previously known range of exponents for spatial dimensions $n \ge
4$.
