The dynamical instability of many-body systems is best characterized through the time-dependent local Lyapunov spectrum [lambda(j)], its associated comoving eigenvectors [delta(j)], and the "global" time-averaged spectrum [<lambda(j)>]. We study the fluctuations of the local spectra as well as the convergence rates and correlation functions associated with the delta vectors as functions of j and system size N. All the number dependences can be described by simple power laws. The various powers depend on the thermodynamic state and force law as well as system dimensionality.