  We introduce two families of ideals, $F$-jumping ideals and $F$-Jacobian
ideals, in order to study the singularities of hypersurfaces in positive
characteristic. Both families are defined using the $D$-modules $M_{\alpha}$
that were introduced by Blickle, Musta\c{t}\u{a} and Smith. Using strong
connections between $F$-jumping ideals and generalized test ideals, we give a
characterization of $F$-jumping numbers for hypersurfaces. Furthermore, we give
an algorithm that determines whether certain numbers are $F$-jumping numbers.
In addition, we use $F$-Jacobian ideals to study intrinsic properties of the
singularities of hypersurfaces. In particular, we give conditions for
$F$-regularity. Moreover, $F$-Jacobian ideals behave similarly to Jacobian
ideals of polynomials. Using techniques developed to study these two new
families of ideals, we provide relations among test ideals, generalized test
ideals, and generalized Lyubeznik numbers for hypersurfaces.
