  The paper addresses the question of existence of a locally self-similar
blow-up for the incompressible Euler equations. Several exclusion results are
proved based on the $L^p$-condition for velocity or vorticity and for a range
of scaling exponents. In particular, in $N$ dimensions if in self-similar
variables $u \in L^p$ and $u \sim \frac{1}{t^{\a/(1+\a)}}$, then the blow-up
does not occur provided $\a >N/2$ or $-1<\a\leq N/p$. This includes the $L^3$
case natural for the Navier-Stokes equations. For $\a = N/2$ we exclude
profiles with an asymptotic power bounds of the form $ |y|^{-N-1+\d} \lesssim
|u(y)| \lesssim |y|^{1-\d}$. Homogeneous near infinity solutions are eliminated
as well except when homogeneity is scaling invariant.
