We study both the longitudinal and transverse fluctuations of a semiflexible filament. Using scaling arguments and numerical simulations, we find several regimes for the the longitudinal fluctuations which for short times scale as $\langle\deltar^2_\|\rangle\sim t^{7/8}$ and are correlated over a length $l_2 \sim t^{1/8}$.  Our results are pertinent to experiments on cross-linked filament systems and motor-filament assays.  The techniques we develop for the analysis of dynamic correlations should have wide applications in the study of polymer systems.