"Use only the Artillery Beam to destroy an enemy ship while taking no hull damage."
This achievement can be easily obtained in Sector 1 by upgrading the shields, then finding an enemy whose weapon volley is either one or two laser shots, or one laser shot and a beam weapon that cannot pierce one layer of shielding. After a while, the the Artillery Beam will finish off the enemy ship and the achievement will be unlocked.
Letting asteroids damage and/or destroy the enemy ship will give the achievement, as long as the other requirements are met. Rockets taken without creating a hull breach on your ship will still allow you to get the achievement.