  We study the problem of $\mathfrak{m}$-adic stability of F-singularities,
that is, whether the property that a quotient of a local ring
$(R,\mathfrak{m})$ by a non-zero divisor $x \in \mathfrak{m}$ has good
F-singularities is preserved in a sufficiently small $\mathfrak{m}$-adic
neighborhood of $x$. We show that $\mathfrak{m}$-adic stability holds for
F-rationality in full generality, and for F-injectivity, F-purity and strong
F-regularity under certain assumptions. We show that strong F-regularity and
F-purity are not stable in general. Moreover, we exhibit strong connections
between stability and deformation phenomena, which hold in great generality.
