The whole development are stuck on one task in a sprint, they would rather work on other tasks then try and solve it. This is the only task remaining in the sprint, all other tasks have finished. The task is not majorly important.
What's the best way to handle this?
Let the team decide. It's their Sprint goal.
It's a short answer because it's a simple problem. Eventually you have to step away from being a Scrum Bouncer and simply let the team make their own way.
They have a goal, they have a task outstanding. Simply watch and observe and take notes on the dynamic then approach the retrospective with the information.
The first step is to understand why the team cannot progress.
What is the challenge? Help them resolve it by removing obstacles, facilitating help/support/knowledge transfer from people or teams.
Encourage the team to complete it as unfinished items in sprints create a habit of leaving things behind. Help the team to stay focused and finish the task.
On the other hand, if however the task is not resolvable, make sure you discuss it in the retrospective
(1) understand why it turned out to be so? For eg, Is this indicate the lack of skills which you need to address? Is it reflective of the lack of planning you have done? Is it the story itself? Is it inappropriate to include such unimportant tasks to sprints which takes up time but add no value?
(2) Agree to the action items/changes you would need to make so that this situation doesn't repeat.
What is the value-add of the task? Developers can smell unnecessary work, and pressing them to do something that few people besides yourself care about can lead to resentment or openly challenging you, especially if the task was selected by you instead of the product owner. If this is a nice-to-have that the PO asked for, and there are other more valuable items in the backlog, you may need to talk with the PO about how/why to order the backlog so the team always works on the most important items first; although even so, the team should be negotiating the scope with the PO, not unilaterally deciding what is and isn't worth doing. If halfway through the sprint they decide not to do something, there should be a valid technical/organizational/capacity reason for dropping the task, not just because they don't feel like it.
If there is genuine value in the task, then you need to make sure there are appropriate consequences if they neglect it. Your first option should be to teach them the importance of keeping their sprint commitment. If that doesn't work, have the PO explain to them why this task is valuable and necessary. If all else fails, escalate to their manager.
