Use the following tips to unlock arenas:
City Cathedral: finish the City Night Showdown Level
Larry's World: finish the Danger Yard Level
Mad Mansion: finish the first part of Jail of Horror Level
Mumbo's Big Top: finish the second part of Magic Mayhem Level
Raven's Mirror: finish the Thriller Driller Level
Titan Tower: finish the Slade Ambush Level
Slade's Lair: finish the Final Clash Level
Sky Needle: finish the Titan Tower Level
The Hive: finish the Hive Academy Level