Instructions
Website
Navigate your Internet browser to a website that checks gamertag availability such as Gamertag Checker or CheckGamertag.com.
Type the gamertag you want to check into the text field on the webpage.
Click the "Check" button. The website will tell you if the name has been taken or not.
Xbox 360
Turn on your console and select your gamertag under the "My Xbox" menu.
Select the "Profile" option, followed by "Edit Profile" and then select "Gamertag."
Select "Enter New Gamertag" and use the virtual keyboard to type in a new gamertag. Press start and you will be told if that gamertag is available or not.