Syntax:
SHOW {DATABASES | SCHEMAS}
    [LIKE 'pattern' | WHERE expr]

SHOW DATABASES lists the databases on the MySQL server host. SHOW
SCHEMAS is a synonym for SHOW DATABASES. The LIKE clause, if present,
indicates which database names to match. The WHERE clause can be given
to select rows using more general conditions, as discussed in
http://dev.mysql.com/doc/refman/5.1/en/extended-show.html.

You see only those databases for which you have some kind of privilege,
unless you have the global SHOW DATABASES privilege. You can also get
this list using the mysqlshow command.

If the server was started with the --skip-show-database option, you
cannot use this statement at all unless you have the SHOW DATABASES
privilege.

SHOW SCHEMAS can also be used.

URL: http://dev.mysql.com/doc/refman/5.1/en/show-databases.html

