SSAS: How to find User-Defined Member Properties of a dimension?

User-Defined Member Properties are attribute relationships added to specific named level in a dimension. Once defined, they can be access using either PROPERTIES key word or Properties function. Here is and example of accessing a user-defined properties.

-- from MSDN
SELECT 
   CROSSJOIN([Ship Date].[Calendar].[Calendar Year].Members, 
             [Measures].[Sales Amount]) ON COLUMNS,
   NON EMPTY Product.Product.MEMBERS
   DIMENSION PROPERTIES 
              Product.Product.[List Price],
              Product.Product.[Dealer Price]  ON ROWS
FROM [Adventure Works]
WHERE ([Date].[Month of Year].[January]) 

However, if you do not know all user-defined properties added and need to get a list, you can simply use Analysis Services Schema Rowsets for querying out information such as objects, state, sessions, connections, etc. There are many schema rowset under OLEDB for OLAP Schema Rowsets and the one we have to use is MDSCHEMA_PROPERTIES. Here is the query for getting all user-defined properties of a level in a dimension.

SELECT *
FROM $SYSTEM.MDSCHEMA_PROPERTIES
WHERE [DIMENSION_UNIQUE_NAME] = '[Dim Customer]'
 AND [HIERARCHY_UNIQUE_NAME] = '[Dim Customer].[Customer]'
 AND [LEVEL_UNIQUE_NAME] = '[Dim Customer].[Customer].[Customer]'
 AND [PROPERTY_TYPE] = 1;

Read more on Analysis Services Schema Rowsets: https://msdn.microsoft.com/en-us/library/ms126233.aspx
Read more on OLE DB for OLAP Schema Rowsets: https://msdn.microsoft.com/en-us/library/ms126079.aspx