If you find a name that is suddenly available, do not hesitate. Go to Settings > Account Management > Account Information > Profile > Online ID. Type it in. If it's free, claim it immediately. There is no "reserve" button.
If you have your heart set on a specific word or a "clan tag," you will likely have to buy it. There is a massive secondary market for gamertags. 4 letter psn names for sale free
Spoiler alert: Every common word (FIRE, GAME, WIN, LOSE, COLD, HOT) is taken. If you find a name that is suddenly