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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., Comment: 29 pages

Let $C\subset \mathbb {P}^n$ be a smooth curve and $N_C$ its normal bundle. $N_C$ satisfies strong interpolation if for all integers $s>0$ and $\lambda _i\in \{0,1,\dots ,n-1\}$, $1\le i \le s$, there are distinct points $P_1,\dots ,P_s\in C$ and linear subspaces $U_i\subseteq E|P_i$ such that $\dim (U_i)= \lambda _i$ for all $i$ and the evaluation map $H^0(E)\to \oplus _{i=1}^{s} U_i$ has maximal rank (A. Atanasios). We prove that $C$ satisfies strong interpolation if either $C$ is a linearly normal elliptic curve or $C$ is a general embedding of degree $d\ge (5n-8)g+2n^2-5n+4$ of a smooth curve $X$ of genus $g\ge 2$.