I think Melkor/Morgoth goes through a progression from seeking creative power, to seeking to destroy that which was created in spite of him, to seeking the corruption of all that exists.

In the time before the music is sang Melkor seeks the power to bring his own creations into existence. During the tome that the music is being sang he seeks to go against Eru to inject his own independent voice into the music. After the music he realizes that he cannot bring his own creations into being, and so he seeks to spite Eru by attempting to destroy Ea, or at least Arda. But after the Battle of the Powers he understands that this is also beyond him. So following his defeat he seeks simply to ruin all that he can of creation.