Is there a way in JIRA to find out (quickly) how many times a task or story has been moved to a certain status?
Not out of the box. The history has it, but it is in line with everything else. I don't know what plug-ins are available in the cloud but one may allow incriminating a counter during a transition and you could add that in for future issues.
On the hosted version of JIRA, it's possible using the script runner plugin and the ChangeHistoryManager API , I don't think you could do it on cloud version of JIRA.
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