What is the best query to use to monitor a SQL Server database's status?
If you're using SQL 2005+ and only want to return the DB name where the DB isn't in the "ONLINE" state I'd use this:

SELECT
    name
FROM sys.databases
WHERE state != 0;


Remember that databases participating in mirroring or log shipping will not be online or may change state regularly. For more info about the sys.databases DMV see documentation here: http://msdn.microsoft.com/en-us/library/ms178534.aspx