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';

You can also query the INFORMATION_SCHEMA.SCHEMATA view:

SELECT SCHEMA_NAME FROM INFORMATION_SCHEMA.SCHEMATA

I believe querying the INFORMATION_SCHEMA views is recommended as they protect you from changes to the underlying sys tables. From the SQL Server 2008 R2 Help:

Information schema views provide an internal, system table-independent view of the SQL Server metadata. Information schema views enable applications to work correctly although significant changes have been made to the underlying system tables. The information schema views included in SQL Server comply with the ISO standard definition for the INFORMATION_SCHEMA.

Ironically, this is immediately preceded by this note:

Some changes have been made to the information schema views that break backward compatibility. These changes are described in the topics for the specific views.

If you are using Sql Server Management Studio, you can obtain a list of all schemas, create your own schema or remove an existing one by browsing to:

Databases - [Your Database] - Security - Schemas

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You can also use the following query to get Schemas for a specific Database user:

select s.schema_id, s.name as schema_name
from sys.schemas s
inner join sys.sysusers u on u.uid = s.principal_id
where u.name='DataBaseUserUserName'
order by s.name

You can execute the following query:

SELECT s.name AS schema_name,
u.name AS schema_owner
FROM sys.schemas s
INNER JOIN sys.sysusers u ON u.uid = s.principal_id
ORDER BY s.name;