Custom Issue Board
Create a board that groups open issues by their owner.
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In Administration > Issue Settings > Fields, add a User field named Owner. Leave Allow Multiple off, enable Allow Empty Value, and name the empty value Unassigned. Enable Include When Issue is Opened for your project. Use a separate field to keep the default multi-valued Assignees field intact.
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Open the project's Issues > Boards, expand the board selector, and click Add New Board. Existing boards can be edited with the pencil icon beside their name.

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Name the board Assignment, set Base Query to
"State" is "Open", select Owner as Identify Field, and choose at least two Board Columns. Include Unassigned and the users whose issues you want to display. Choose Display Fields for the cards, then save.
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Select Assignment. With Unscheduled selected, the board shows matching issues that have no iteration. Choose an iteration to see its issues instead. Existing issues may need a batch edit to populate the new Owner field.

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Drag an issue by its dotted handle into another column to change its Owner. Moving it into Unassigned clears Owner. This requires permission to edit that issue field.