The following is the description of the Oracle error you are receiving:
"When ORA-00904 occurs, you must enter a valid column name as it is either missing or the one entered is invalid. The "invalid identifier" most common happens when you are referencing an invalid alias in a select statement. " (http://www.dba-oracle.com/t_ora_00904_string_invalid_identifier.htm)
Perhaps one of your custom forms refers to a field that has changed name or been removed?
At what point do you get this error ? During the upgrade ? When you try to log in for the first time afterwards ? Accessing a particular form ?
You should check the log files from the upgrade - it should tell you if something didn't work as expected.
It seems something is wrong in the SDE Forms in your database and hence the issue.
Your need to contact BMC SDE support for further help on this issue.
In responding to your note. We get this error when we try to log into SDE for the first time afterwards.
Did you have to make any changes to the database as part of the upgrade ? Any field deletions / renames ?
Unfortunately I don't know which log file might help in this, but it may be worth running a sql trace as it may tell you which form is the problem. Get your oracle DBA to help you with this.