The authors first extend the sharp Wolff-type decoupling estimates of J. Bourgain and C. Demeter [Ann. Math. (2) 182, No. 1, 351–389 (2015; Zbl 1322.42014); J. Anal. Math. 133, 279–311 (2017; Zbl 1384.42016)] from the constant coefficient case to the variable coefficient setting. Then they apply these results to obtain some new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, they also establish local smoothing estimates for a natural class of Fourier integral operators. It should be pointed out that at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.