When messages are missing = POP issue
Are you missing messages when you sign into your account at mail.aol.com? Check the POP settings in whatever other program you use to check your AOL Mail (Outlook, Mac Mail, Entourage, etc.; and mail on your mobile).
In that program's POP settings, make sure "Leave messages on server" is selected. That keeps the messages in your AOL inbox as well as importing them to wherever else you view them.
For Outlook, here's what that looks like. You can find this option on the Advanced tab of 'Internet E-mail Settings' after clicking More Settings... in the Add New E-mail Account tool:
In whatever program you're using, check the box to "Leave a copy of messages on the server."
When messages are 'Read' or show up in 'Old', 'Recently Deleted' or 'Trash' = IMAP issue
Do messages mysteriously appear as 'Read', or go straight into Old, Recently Deleted or Trash? You probably check your AOL mail using another program (Outlook, Mac Mail, Gmail, etc.) ... or perhaps from another device (the mail client on your mobile phone, for example). True? If so, then this is an IMAP issue.
IMAP syncs a mail account across different devices and programs. So when you read one of your AOL Mail messages on your mobile, or in Outlook, the message then shows up as 'read' everywhere else. That's why it would appear in your Old folder when you sign into mail.aol.com. And with IMAP, if you delete a message while checking mail on your mobile device or somewhere else, it's also deleted when you sign in at mail.aol.com. It will be in Trash.
POP vs. IMAP
POP pulls your mail -- downloads it -- into the application where you read it (Outlook, Mac Mail, etc.) That means any new messages get saved wherever you download them (in Outlook, Mac Mail, etc.). Then, unless you've selected "leave a copy of messages on the server", POP deletes the messages from their original location. In this case, that's the AOL Mail servers.
IMAP doesn't do this. IMAP synchronizes the read/unread status of your mail and leaves it in both places -- in the email application where you are reading your mail (Outlook, Mac Mail, etc.) as well as at mail.aol.com. If a message is read in either place, it appears as "Read" in both places. If it's unread, it appears as "Unread" in both places. If a message is moved to Trash, it's moved to Trash both places. Etc.