Usefulness of a linked list with a cycle
In the case of circular linked list - this might be a buffer, neat structure to handle operations that must be consecutive or form a ring (like in parallel programming).

What if not the whole list is cycle?
This might be intended like conflict resolving, and thanks to cycle it will not go further.
It might be the case that every operation changes state so there is cycle, but it will eventually resolve.  

If you write a turn-based game with general loop, there might be cycle if user can do something multiple times per turn. So here it is intended.

And totally not constructive - cycle came from bug, so it pushes application into infinite computation.