The access to the Museum is limited. At first exploring it is completely impossible, as the a laser barrier is protecting access to the main rooms. It changes after completing the Disable Penguin's Final Communications Disruptor underground main mission.
The museum has two entrances. The more obvious option implies using the front door (screen above), beside which there's always a small group of enemies. Choosing this entrance guarantees access to most of the Museum rooms. During your first visit though, you will have to use the Cryptographic Sequencer to unlock the door.
The only exception is the Iceberg Lounge. Exploring the room is possible only after completing the Defeat Penguin main mission. You unfortunately can't get there by choosing the path leading to the torture chamber and as a result you will have to locate the northern Museum entrance (screen above). Just like with the front door, you will have to defeat a small group of inmates.
Batman Trophies - 23
Riddles - 8
Penguins - 12 (4 challenges)
Catwoman Trophies - 5