  In this paper, we prove that the normal bundle of a general Brill-Noether
space curve of degree $d$ and genus $g \geq 2$ is stable if and only if $(d,g)
\not\in \{ (5,2), (6,4) \}$. When $g\leq1$ and the characteristic of the ground
field is zero, it is classical that the normal bundle is strictly semistable.
We show that this fails in characteristic $2$ for all rational curves of even
degree.
