Center Perks - Prison Takeover (If you do this with nunchucks, you knock out another achievement.)
Raise your strength and speed to at least 80 each, I'd recommend strength to 90 for the extra health. Get at least 3 duct tapes/ropes, and begin to beat up guards. Tie up three and continue your spree. The warden will open the front gate. Leave through it.
Stalag Flucht - prison takeover
Same method, one more step. You need to make a copy of the red key to get out. (Wad of putty, molten plastic.) Combine the putty and the key, then put the key back. You will have a mold that you combine with the plastic to get a plastic key. Once the gate is opened from your takeover, use the key to get through the only door left in the way of freedom.
Shankton State Pen - Digging.
This prison has 10 guards and 20 inmates. I found the best way out of here was to dig.
There is, conveniently, a hallway behind your cell perfect for digging and storing materials. You can break through this wall using Plastic Forks OR a Pickaxe. You will also need to craft a Shovel of whatever grade you desire (obviously the Sturdy Shovel will work the best.) and a Poster OR Fake Wall Block to cover your trail. And begin to dig your way through the corridor behind your cell.
After digging a few blocks, you will get a notification telling you to support the tunnel before digging any further. You will need to craft a Timber Brace to do this. BE AWARE that lots of these will be needed, and so a job in the Wood Shop will serve you well (alternatively, timber ((as well as metal sheets for the shovels, if you don't fancy a job in Metal Works, either)) can be bought from your fellow inmates). In order to smuggle Timber out of the Wood Shop, you must craft a Contraband Pouch. (Simply hold it in your inventory with any Contraband, and you won't be detected.)
Once you have dug until the game will let you dig no further, get yourself a Guard Outfit, either from beating up a guard and stealing one, crafting one, or buying one from a fellow inmate, pop it on and book it out of there.
Jungle Compound - Prison takeover.
You need a few more things for this one - a feather, ink and the red key again. Head north when the takeover is initiated, go to the room with the unsigned papers in and combine them with your feather and ink to get signed papers. Then all the way to the south is the guard checkpoint where you can present the papers and leave.
San Pancho - Digging.
This is easy yet time consuming. Make friends with everyone in your cell to prevent snitching. Then move the far left desk, dig in its place and put it back. Underground, you will need to go a full 24 spaces left. Avoid mines (you can see them in the ground BEFORE you run into them, so dig around them or game over) and dig around boulders. Easy, but it depends on your luck with getting the items required to dig this far (you can always use spoons for a little bit more progress)
HMP Irongate - Prison takeover.
I've never escaped this admittedly, but the guys over on Xbox have made a pretty sound method to getting it done fast and efficient. I'm going to copy paste it here - bear in mind a few things are different across versions, but you should be able to figure it out.
(credit to Trueachievements member jodiebears on this solution)
This prison has 10 guards and 17 inmates. Although this is the hardest prison of the lot, we're going to beat our way out. (I managed to escape on day 6 using this method.)
Now, a few differences on this prison:
There are metal/contraband detectors EVERYWHERE. You could craft a Contraband Pouch, but I found the easiest way to get Contraband materials was to make whichever inmate you wish to buy from your friend, have him follow you to your cell, and then buy what you want in there and send him packing. This way you avoid the detectors completely, and save a bit of hassle.
The job you have doesn't much matter here, I chose the Wood Shop as it seems (to me, at least) one of the quickest jobs to have. This means quick money, but in addition to this, you'll want to complete as many favours as you possibly can (I found the "cause a ruckus at whatever" to be the easiest/most profitable. Remember refusing favours causes a small impact on friendship with the selected inmate, but it's easily rectified through presents.)
Now, the guards here are tougher than the previous prisons. You will need to craft either Nunchucks (which CAN also be bought from fellow inmates, on rare occasions) or a Whip (I chose the whip) as they do the most damage. Our trusty Plated Inmate Outfit will do just fine for defence when coupled with Medicine, which can be bought from your fellow inmates.
You will also need to craft a Make Shift Raft using Balsa Wood, an item only found on this level. Since this prison is on an island, this is your ONLY means of escape.
In addition to this, we will also need a Red Plastic Key, so be sure to craft some Putty and Molten Plastic in order to make this. (I started with the necessary materials to make both of these in my desk. If, however, you do not, you can always buy the materials from fellow inmates, or beat people up and take them.)
Buy your Balsa and Timber wood from your fellow inmates, as breaking into cells and stealing from desks gets the guards on your tail pretty fast as each cell in this prison has it's own security camera. Take your time, and build up your stats as far as you can.
Once you're ready for escape, simply equip your Plated Inmate Outfit and your weapon, have your Makeshift Raft, Putty, Molten Plastic, and whatever Medicine you can carry with you, and set off on your killing spree. Try to gather the guards into groups before beating them down, as otherwise they will spread out and wake up before Lockdown is initiated, making this harder on you. I would recommend having everything ready, and waiting until morning (meaning you have a fresh save to load if anything goes wrong) before attempting.
Find the guard with the Red Key, make yourself a Plastic Red Key, and keep the punches rolling until lockdown is in effect. Once you have the notification, run South (Beware the electric fence) and run to the dock on the right of the island. Simply hover over the Makeshift Raft in your inventory, and you'll sail away. Yo ho ho and a bottle of rum. Tastes like freedom!