In the Main Cell Block:
"This room is the end of days for even the most celebrated killer.", hmmm - now where could this be? You simply need to go to the Calendar Man's cell in this level to complete this riddle. Don't know which it is? Easy, look for the cell with all of the calendar pages covering it.
Next, you'll get "Two people, one voice, no gun?" after you fight Harley's mobs that are on top of all the cells. You need to look for Scarface's gun (a tommy gun of course) which is in a display case that's between the two different control rooms in the Pen.
In the Cell Access area:
"How to you reflect on your successes and failures, Batman?", yes, another odd riddle. Simply look at the mirror in the women's bathroom of all places and scan it. You'll complete the riddle there.
In the Guard Room:
"When is something right in front of you but still hidden from view?", you will need the Zipline and the Bat Claw (that is upgraded) for this riddle. Simply clear out the gas that is in this area by climbing up into the vents that are above where the guards are. Then, bust through that breakable wall that is there and use your Bat Claw to then pull the other wall down. Then, take out your Batarang and hit the breaker to turn the fans on for that area. Next, use your zipline to go across this area and then when you drop, turn around, turn on Detective mode and look back up where you just were at.
Next, you'll hear "Prometheus, Arkham guards' most wanted and most hated." Simply drop down after you've cleared the gas and check out the message board in that area. Then, zoom in on the Prometheus posting. Riddle solved.
In Extreme Incarceration area:
"All alone in your cell? Why don't you break the ice with the most dangerous prisoners?". This one is actually really easy to do. Simply defeat Harley and lock her up, then find the cell that is covered in ice.
In Security Control Room:
"A case of mistaken identity?" is the small riddle here, and you just have to find and scan the cell that has no skeleton in it. Make sure that you're in detective mode when you do this, and you'll find Clayface this way.
In Controlled Access:
"What has four walls, two sides and one ex-DA?", now, for any Batman fan, you should easily be able to figure out who this one is about! You just need to find Two-Face's cell, and it's so easy as it has a ton of Harvey Dent posters all over it.