  We prove that for every smooth compact manifold $M$ and any $r \ge 1$,
whenever there is an open domain in $\mathrm{Diff}^r(M)$ exhibiting a
persistent homoclinic tangency related to a basic set with a sectionally
dissipative periodic saddle, topologically generic diffeomorphisms in this
domain have Lyapunov unstable Milnor attractors. This implies, in particular,
that the instability of Milnor attractors is locally topologically generic in
$C^1$ if $\mathrm{dim}\,M \ge 3$ and in $C^2$ if $\mathrm{dim}\,M = 2$.
Moreover, it follows from the results of C. Bonatti, L. J. D\'iaz and E. R.
Pujals that, for a $C^1$ topologically generic diffeomorphism of a closed
manifold, either any homoclinic class admits some dominated splitting, or this
diffeomorphism has an unstable Milnor attractor, or the inverse diffeomorphism
has an unstable Milnor attractor. The same results hold for statistical and
minimal attractors.
