Rock It Trophy

This is cumulative across both Challenge and Arcade mode. There will of course be plenty of Asteroids to shoot and kill throughout the game (it's only called Asteroids, afterall). As soon as you've "killed" your 1,000th Asteroid, the trophy will unlock. 

If you want to grind this out instead of waiting for it to unlock naturally, you can replay Challenge #2, "Final Burst", repeatedly until you get it. This Challenge has a lot of Asteroids to "kill" with the triple and x7 spread shots, allowing you take out more at one time versus if you just use your single shot/other power-ups.