First-ever tip article! Woo! Check back tomorrow for a map-specific tip for Arkham City's Top of the World Extreme challenge map.
This is actually not one tip, but small collection of tips for combat challenge maps. Some of these are from my gameplay, and others are from friends' observations.
When playing as Batman, if you have the skins DLC, use the Batman Beyond costume. Since it doesn't have a cape to obscure things, you can see what's going on around you just a little bit more clearly.
When playing as Batman, make use of your disarm and destroy combo as soon as you can and as often as you can. It's best not to have to worry about those pesky knives and riot shields. However, none of Catwoman, Robin, and Nightwing have access to the disarm and destroy combo. As these characters, make sure to jump around a lot -- since knives and shields can't be countered, it's best to just avoid them.
In combat challenges, the most important thing is to build up your combo. Using combo takedowns is actually worse for your score than racking up a higher combo count. Don't even counter -- jump over attackers and then strike for the 2x critical combo bonus rather than the 1x counter bonus.
The highest-scoring move you can access normally is the ground pound. Always finish off the last thug with a ground pound by using a super stun for a single-move score of 75x your current combo count. If there's a Titan thug, try to save him for last for a 100x finisher.