How do I obtain a list of all schemas in a Sql Server database

For 2005 and later, these will both give what you're looking for.

SELECT name FROM sys.schemas
SELECT SCHEMA_NAME FROM INFORMATION_SCHEMA.SCHEMATA

For 2000, this will give a list of the databases in the instance.

SELECT * FROM INFORMATION_SCHEMA.SCHEMATA

That's the "backward incompatability" noted in @Adrift's answer.

In SQL Server 2000 (and lower), there aren't really "schemas" as such, although you can use roles as namespaces in a similar way. In that case, this may be the closest equivalent.

SELECT * FROM sysusers WHERE gid <> 0

Try this query here:

SELECT * FROM sys.schemas

This will give you the name and schema_id for all defines schemas in the database you execute this in.

I don't really know what you mean by querying the "schema API" - these sys. catalog views (in the sys schema) are your best bet for any system information about databases and objects in those databases.

SELECT s.name + '.' + ao.name
       , s.name
FROM sys.all_objects ao
INNER JOIN sys.schemas s ON s.schema_id = ao.schema_id
WHERE ao.type='u';