How can I list all user-owned functions?

You can list any user owned objects by querying the user_objects table.
To do this for functions:

select object_name from user_objects where object_type = 'FUNCTION';

This table can also show other objects, like procedures and packages.
The user_objects table is supplemented by the table all_objects, which contains all the objects the current user has access to (i.e. grants to tables in other schemas etc).
The object_types this can be queried for on both tables is as follows:

select distinct object_type from all_objects;

OBJECT_TYPE
-------------------
JOB CLASS
INDEXTYPE
PROCEDURE
JAVA CLASS
SCHEDULE
WINDOW
WINDOW GROUP
JAVA RESOURCE
TABLE
TYPE
VIEW
FUNCTION
PROGRAM
SYNONYM
CONSUMER GROUP
EVALUATION CONTEXT
OPERATOR
PACKAGE
SEQUENCE
XML SCHEMA

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