These are some helpful walkthroughs for some side missions that occur during the aftermath of Arkham City's main story.
Catwoman Mission Walkthrough - Part #1
In my opinion, Catwoman is the most boring character in Arkham City. Some stuff about her I like, but aside from her ability to walk on areas of the ceiling, she is probably the worst part of the game. Her thief vision is like a thermal view but is really vomit inducing. So I'll try my best to explain good ways to use her full potential.
The mission starts off with Catwoman saying she's going to get her stuff and leave this "wretched place." Like previously, enter the cage area where you went to retrieve your stuff early on in the game. Then you learn that bombs have been planted in there. Fight of some of Two-Face's thugs and interrogate them afterward.
Catwoman's health drains really quickly, so spam the X button and move around the analog stick and she will take each one out in turn really fast. I guess that's one thing she can do better than Batman. Do not bother with whip unless having to fight enemies with riot shields.
-Circle(Whip)+X+X= Aerial Attack
After interrogating the thug, you learn Two-Face is in the Museum all the way across the map from where you are. Try not to engage in combat and use your inefficent whip to move to the museum.
Once inside, head to the end of the hallway in front of the moving T. Rex and turn left. Some of the goons will be talking about you when you enter so don't engage just yet. Get up onto the pterodactyl and from there pounce on one of the goons. Beat them all up and go down the stairs. Keep running to the end of the hallway and open the door. Follow the arrow in the red paint detailing, "New Recruits This way!" Keep running to the end of the hallway and turn left. Then slide underneath the metal fence/door leading to the gladiator pit. (A+RT/X+RT)
Take the thugs in there. Use the strategy described earlier, only use the whip when needed for riot gear. Otherwise, use the X/Square button while flicking the analog stick.
Catwoman Mission Walkthrough - Part #2
After defeating the thugs, go onto the right side of the huge fence surrounding the area and open the door. Walk up the curved staircase and into the door at the end of the hall. The ice is completely gone from earlier, but Catwoman is able to go underneath the platform. Crawl underneath, upside down the platform and make your way to the Armory. You can pick up 2 riddler trophies along the way. Inside the Armory, Two-Face and his men have situated them selves. Two-Face and a sniper are in the center, and guards circle around the upstairs and downstairs. Crawl onto the designated ceiling area, and proceed to silent take-downing everyone. A man should come underneath you along the way, and takedown him. Crouching, or crawling (Hold down RT) makes you practically invisible to people, but don't test it right in front of them. Get both of the people upstairs (not including Two-Face and his right-hand man. They alternate spots about every 40 seconds, so go to the farside of the room and down the stairs. Two men are situated down here, and silent takedown one of them while the other is oblivious. Then him as well. Go back upstairs, and when Two-Face's back is turned, silent takedown his guard. Then proceed to beat Two-Face up. Two-face is really easy, and he may not even land a hit on you if you're fast enough. Then he reveals he sold half of your stuff. Afterwards, you can start playing as Batman again.