I have a university email configured using POP on iPhone XS running iOS 13.1.2. Even though I have sent email on the phone, the sent folder is missing in the list of Mailboxes. Sent email will disappear and you cannot find it even through search after closing and reopen the app.
This is also reported in the Apple community: https://discussions.apple.com/thread/250646081
Here is a potential solution I have found accidentally for the missing sent folder issue in iOS 13.
1. Send a email using your iPhone. I send it to my other email account so I can make sure the email was actually sent.
2. Search the address you sent to in the mail app, the email you just sent should show up in the search result;*
3. From the search result, delete the email you just sent.
4. Go back to the list of mailboxes, sent folder should show up. The delete email should be in the trash folder.
*If you close the mail app after sending the email, open mail again, and search, the search will not find the email you just sent. In this case, send a second email to the same address and search the address while mail app is open, both email will show up.
Post here if this solves the problem on your phone. I have only tried it on my own phone and am not sure if it is a general solution. I hope the next update of iOS fixes this issue.
This is also reported in the Apple community: https://discussions.apple.com/thread/250646081
Here is a potential solution I have found accidentally for the missing sent folder issue in iOS 13.
1. Send a email using your iPhone. I send it to my other email account so I can make sure the email was actually sent.
2. Search the address you sent to in the mail app, the email you just sent should show up in the search result;*
3. From the search result, delete the email you just sent.
4. Go back to the list of mailboxes, sent folder should show up. The delete email should be in the trash folder.
*If you close the mail app after sending the email, open mail again, and search, the search will not find the email you just sent. In this case, send a second email to the same address and search the address while mail app is open, both email will show up.
Post here if this solves the problem on your phone. I have only tried it on my own phone and am not sure if it is a general solution. I hope the next update of iOS fixes this issue.