Re: I'm sorry but I have a *SERIOUS* bone to pick with this game(DLC related)!!!!
It seems the only way to cause significant damage to an opponent's car is to t-bone them. That means to hit the side of an opponent's car with the front of your car. Doing this as fast as possible is better, and using boost helps. Classic Carnage events have boost enabled in the races, but you'll have to remember to turn that option on in custom races in the race creator.
Opponent name indicators are important for demolition events, since name indicators have health indicators below them. Pay attention to them to see what damage the cars already have, or if they're already wrecked, since sometimes it's impossible to tell if a car is already wrecked without looking at the health indicator. If an opponent is already wrecked, you're wasting time hitting them. For causing points of damage it's ok to hit any car that's not already wrecked, but for goals involving wrecking opponents, only aim for cars that are low on health. It takes too long to wreck a healthy car if you have to wreck multiple cars in a single race. Drive around and keep looking at health indicators, and eventually you'll see one that's low, then wreck that car.
Until I figured this out by trial and error, I was frustrated. After I figured this out, I hated demolition derby a lot less than drift events.