Is the language Turing machine-Input pairs for which the computation loops recognizable?
The language $L$ is recognizable: you can build a Turing machine that recognizes $L$ by simulating $M$ and keeping track of all the traversed states. As soon as a state is repeated, you accept. If $M$ halts and no stated is repeated, then reject.
Moreover, $L$ is not decidable. Suppose that $L$ was decidable, then you could do the following to solve the halting problem:

Create a new Turing machine $M'$ behaves as $M$, but instead accepting/rejecting, it goes into some looping state.

Simulate a Turing machine $T$ that decides $L$ to check whether $\langle M', w \rangle \in L$.

If $\langle M', w \rangle \not\in L$ then $\langle M, w \rangle$ must not halt.

If $\langle M', w \rangle \in L$ then either $\langle M, w \rangle$ halts or loops.

Simulate $T$ to check whether $\langle M, w \rangle \in L$.

If $\langle M, w \rangle \in L$ then $\langle M, w \rangle$ loops, and hence it does not halt.

If $\langle M, w \rangle \not\in L$  then $\langle M, w \rangle$ does not loop, therefore it halts.