Here's one I haven't seen a discussion about.
A client has a website. He wanted a short URL, e.g., mysite.com, so he wouldn't have to constantly tell people that his website address was MyVeryLongAndComplicatedBusinessName.com. It's set up so that the site itself is located at mysite.com, with the very long and complicated name pointed to it.
His formal business materials use the long name, but when he is on the phone or in conversation he tells people that his site is mysite.com. His main email address is firstname.lastname@example.org.
Now he's beginning to be concerned because he has no real page ranking for mysite.com, and one that is lower than he would like for MyVeryLongAndComplicatedBusinessName.com.
Is the dual naming causing a problem? If so, should we swap the domain names so that mysite.com is a pointer to to actual web space at MyVeryLongAndComplicatedBusinessName.com?