I found Syncopia to be the easiest course to get the Detonator and Zone Battle trophies on as its wide, theres no splits and there isnt really any nasty corners.

If you fail first time with Hold Your Fire, try again as the pattern the mines are spread seem to change every game so next time they may be more manageable. For Da Bomb, it may be worth saving your EMP specifically to take the bomb out, so you dont miss it if its not in your firing/racing line.

For the sharpshooter trophy, wait a few laps so the mines have built up a bit before blasting away, and make sure you pick a big group to fire at. Make sure you tap when you press the fire button, not hold. As your bullets keep going through a mine when you take it out, if your rapid firing some bullets will most likely miss. In fact this is a good tip generally as youll find you take a lot more mines firing this way.