  Let $X$ be a nonsingular complex projective toric variety. We address the
question of semi-stability as well as stability for the tangent bundle $T{X}$.
In particular, a complete answer is given when $X$ is a Fano toric variety of
dimension four with Picard number at most two, complementing earlier work of
Nakagawa. We also give an infinite set of examples of Fano toric varieties for
which $TX$ is unstable; the dimensions of this collection of varieties are
unbounded. Our method is based on the equivariant approach initiated by
Klyachko and developed further by Perling and Kool.
