The easiest way to destroy a Quark Mine to use a projectile (preferably controlable) from a distance to blow it up. Using a counterattacking move is also effective, as you don't get hit by the Quark Mine while you use the move. In this case, Toad is the most effective counter move for this job, since the spores the Toad releases can blow up other Quark Mines in the process. Most Quark mines are found on the Isle of Ancients, in the Research Facility, and in the Subspace Bomb Factory, while others can be found in The Wilds, on the Halberd, and throughout The Great Maze.