Hello, @Philipp B,
Welcome to the Microsoft Q&A platform!
In order to resolve your issue, you can try some common troubleshooting like rebuilding the search index, ensuring your Exchange Server is updated, applying a registry fix to disable FAST search, restarting the Exchange search services, and considering moving mailboxes to another database.
Here are some similar issues and relative documents for your reference:
If all of these methods fail, according to Service health in the M365 Admin Center, the issue regarding “Users' searches may fail and provide no results in the Outlook desktop client and Outlook on the web.” has been listed as an unresolved issue that Microsoft is working on, which means that the problem you are experiencing is a global issue.
Now the development of code fix to trigger automated recovery service to restart infrastructure supporting search functionality and clear any errors is complete. Microsoft are preparing deployment procedures to implement the fix by the end of January 2025 and will closely monitor the progress, just as the screenshots shows below.
It is recommended for you to keep monitor this known issue in Microsoft 365 Health Portal for the next update if you have access to the M365 Admin Center. If not, I will update the thread here once I get additional information and the specific timeline for the Service Incident once fixed.
Should you need more help on this, you can feel free to post back.
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Thank you for your support and understanding.
Best Wishes,
Alex Zhang