  We provide a uniform approach to obtain sufficient criteria for a (higher
order) fixed point of a given bracket structure on a manifold to be stable
under deformations. Examples of bracket structures include Lie algebroids, Lie
$n$-algebroids, singular foliations, Lie bialgebroids, Courant algebroids and
Dirac structures in split Courant algebroids admitting a Dirac complement. We
show that the stability problems are specific instances of the following
problem: given a differential graded Lie algebra $\mathfrak g$, a differential
graded Lie subalgebra $\mathfrak h$ of degreewise finite codimension in
$\mathfrak g$ and a Maurer-Cartan element $Q\in \mathfrak h^1$, when are
Maurer-Cartan elements near $Q$ in $\mathfrak g$ gauge equivalent to elements
of $\mathfrak h^1$? We show that the vanishing of a finite-dimensional
cohomology group associated to $\mathfrak g,\mathfrak h$ and $Q$ implies a
positive answer to the question above, and therefore implies stability of fixed
points of the geometric structures described above. In particular, we recover
the stability results of Crainic-Fernandes for zero-dimensional leaves, as well
as the stability results for higher order singularities of Dufour-Wade.
