How to use a variable for the database name in T-SQL?

Put the entire script into a template string, with {SERVERNAME} placeholders. Then edit the string using:

SET @SQL_SCRIPT = REPLACE(@TEMPLATE, '{SERVERNAME}', @DBNAME)

and then run it with

EXECUTE (@SQL_SCRIPT)

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It's hard to believe that, in the course of three years, nobody noticed that my code doesn't work!

You can't EXEC multiple batches. GO is a batch separator, not a T-SQL statement. It's necessary to build three separate strings, and then to EXEC each one after substitution.

I suppose one could do something "clever" by breaking the single template string into multiple rows by splitting on GO; I've done that in ADO.NET code.

And where did I get the word "SERVERNAME" from?

Here's some code that I just tested (and which works):

DECLARE @DBNAME VARCHAR(255)
SET @DBNAME = 'TestDB'

DECLARE @CREATE_TEMPLATE VARCHAR(MAX)
DECLARE @COMPAT_TEMPLATE VARCHAR(MAX)
DECLARE @RECOVERY_TEMPLATE VARCHAR(MAX)

SET @CREATE_TEMPLATE = 'CREATE DATABASE {DBNAME}'
SET @COMPAT_TEMPLATE='ALTER DATABASE {DBNAME} SET COMPATIBILITY_LEVEL = 90'
SET @RECOVERY_TEMPLATE='ALTER DATABASE {DBNAME} SET RECOVERY SIMPLE'

DECLARE @SQL_SCRIPT VARCHAR(MAX)

SET @SQL_SCRIPT = REPLACE(@CREATE_TEMPLATE, '{DBNAME}', @DBNAME)
EXECUTE (@SQL_SCRIPT)

SET @SQL_SCRIPT = REPLACE(@COMPAT_TEMPLATE, '{DBNAME}', @DBNAME)
EXECUTE (@SQL_SCRIPT)

SET @SQL_SCRIPT = REPLACE(@RECOVERY_TEMPLATE, '{DBNAME}', @DBNAME)
EXECUTE (@SQL_SCRIPT)

You can also use sqlcmd mode for this (enable this on the "Query" menu in Management Studio).

:setvar dbname "TEST" 

CREATE DATABASE $(dbname)
GO
ALTER DATABASE $(dbname) SET COMPATIBILITY_LEVEL = 90
GO
ALTER DATABASE $(dbname) SET RECOVERY SIMPLE 
GO

EDIT:

Check this MSDN article to set parameters via the SQLCMD tool.