Functional Data Analysis (FDA) depends critically on the regularity of the observed curves or surfaces. Estimating this regularity is a difficult problem in nonparametric statistics. In FDA, however, it is much easier due to the replication nature of the data. After introducing the concept of local regularity for functional data, we provide user-friendly nonparametric methods for investigating it, for which we derive non-asymptotic concentration results. The results are obtained under weak dependence conditions between the curves. Usual functional time series models (functional autoregressive, functional ARCH, etc) satisfy our conditions. As an application of the local regularity estimation, we propose adaptive estimators for the mean and autocovariance functions. Extensive simulation experiments illustrate the performance of our estimators with finite series.