I wrote an article to List SQL Server read/write permissions, and a user commented it would be nice to get something similar about Azure SQL Database. The query is basically the same as for a single database, but there are several things you need to be aware of:
In Azure, when you create the database you must create a server, there you specify the administrator user and password. This will be a server login with permissions in the [master] database which is able to create other server logins. You can view the server logins querying [sys].[sql_logins], there is no table