  To explain the accelerated expansion of our universe, many dark energy models
and modified gravity theories have been proposed so far. It is argued in the
literature that they are difficult to be distinguished on the cosmological
scales. Therefore, it is well motivated to consider the relevant astrophysical
phenomena on (or below) the galactic scales. In this work, we study the
stability of self-gravitating differentially rotating galactic disks in $f(T)$
theory, and obtain the local stability criteria in $f(T)$ theory, which are
valid for all $f(T)$ theories satisfying $f(T=0)=0$ and $f_T (T=0)\not=0$, if
the adiabatic approximation and the weak field limit are considered. The
information of the function $f(T)$ is mainly encoded in the parameter
$\alpha\equiv 1/f_T(T=0)$. We find that the local stability criteria in $f(T)$
theory are quite different from the ones in Newtonian gravity, general
relativity, and other modified gravity theories such as $f(R)$ theory. We
consider that this might be a possible hint to distinguish $f(T)$ theory from
general relativity and other modified gravity theories on (or below) the
galactic scales.
