  We study blow-up rates and the blow-up profiles of possible asymptotically
self-similar singularities of the 3D Euler equations, where the sense of
convergence and self-similarity are considered in various sense. We extend much
further, in particular, the previous nonexistence results of
self-similar/asymptotically self-similar singularities obtained in
\cite{cha1,cha2}. Some implications the notions for the 3D Navier-Stokes
equations are also deduced. Generalization of the self-similar transforms is
also considered, and by appropriate choice of the transform we obtain new
\textit{a priori} estimates for the 3D Euler and the Navier-Stokes equations.
