Why is $L = \{ \langle M \rangle : L(M) = \{ \langle M \rangle \} \}$ not Turing-recognizable?
Given a Turing machine $T$, use the recursion theorem to construct a Turing machine $T'$ that halts if its input $x$ is $\langle T' \rangle$, runs $T$ on $x$ if $x &lt; \langle T' \rangle$, and runs $T$ on $x-1$ if $x &gt; \langle T' \rangle$. If you could recursively enumerate $L$, then you could use this reduction to recursively enumerate all machines that never halt.