Challenge Mode Trophies
The challenge trophies, are definitively the hardest in my opinion. Difficulty does effect score, however, a perfect run on Normal Difficulty is easier than a good run on Impossible Difficulty.
The biggest key is kill streaks. Kill streaks are kills you have gotten in a row WITHOUT receiving damage.
Making sure to keep enemies at bay is crucial. The contact beam, force gun, and line gun are great for this mode.
Overtime Sheet 
Beat the top score on the "Extraction Site" Challenge
251,010 points
In this level, you will be fighting through the preparation area, all the way to to the first door beyond the marker
No Cop Duty
Beat the top score on the "Central Plaza" Challenge
252,010 points
In this level, you will fight from P-Sec Headquarters' front door to the door to the left of the entrance
Engineer By Trade
Beat the top score on the "Maintenance Bay" Challenge
253,010 points
You will be fighting in the area where you took off in the shuttle from the colony. Quick tip. Babies and Pregnants should be taken down with the Contact Beam
Laugh It Up Partner
Beat the top score on the "Mess Hall" Challenge
254,010 points
You'll start on the top floor of the mess hall, and work your way down to the lower exit. We are introduced to spitters here. Take away their power with head shots
Colony Full Of Crazy People
Beat the top score on the "Medical Bay" Challenge
255,010 points
You start near the security depot, and take several laps around the main ER room. Force gun away!
Need A Towel
Beat the top score on the "Water Tunnels" Challenge
256,010 points
You trek through the knee-high water. Really, just force gun, and make sure to blast the exploders from afar.
Herbal Remedies
Beat the top score on the "East Grow Chamber" Challenge
257,010 points
You will take several laps around the East Grow Chambers. Stasis the flying necromorphs, and shoot them down with the rivet gun when nothing else is on the screen.
Ready For Takeoff
Beat the top score on the "Shuttle Bay 47" Challenge
258,010 points
The force gun is your friend here, as you battle around the shuttle you use to escape in the story.
Bridge Commander
Beat the top score on the "Atrium" Challenge
259,010 points
The force gun is crucial here, especially when near the captain's nest, as the necromorphs come from both sides.
Get A Medic
Beat the top score on the "Life Support Room" Challenge
260,010 points
We go back to the first chapter's life support room. The twist on the rest of the challenges is in between each round, you can wait as long as you like. This means take the time to let your stasis recharge, and reload all your guns. Enjoy the force gun on all the leapers, and switch to your stasis and rivet gun (or contact beam if you get it luckily).
Action Trophies
You Need Glasses Partner?
Survive the shuttle flight with less than 10% shields intact
On chapter 4, you will be firing a cannon. I would advise to defend as well as you can against the asteroids. The ones you need to worry about are the gold ones. Once you get the cannon fire you can let most of them hit you. They take just a little over 5% for each group of 4 on Hard. You will have to defend one or two waves of the cannon fire near the end, but it should be easy to predict.
Twice The Firepower
Play cooperatively with a second player
This will unlock the instant a second player joins in. This can be in the middle of a chapter, beginning, or whenever. I obtained with two dualshock controllers, but any combination of move and/or standard controllers should work.
Sharpshooter
Shoot the box hanging from the crane on the surface of Aegis 7
In chapter one, right after the first bridge before the Marker, there is a crane with a box. Shoot it down by aiming at the cord right above it.
Too Much Firepower?
Open the secret weapons store in P-Sec
After Lexine joins your party in chapter 2, and the go down the elevator, quickly shoot the fuse box to access the supply closet.
What Did You Find?
Take a minor detour in the megavents
In chapter 3, when you have a choice to go down a ladder or cut through some "meat", choose to cut through the meat. You'll eventually come to a supply closet.
Taking Your Time, McNeill?
Admire the view in the crashed shuttle
Seriously? Ok. On level 4, after your shuttle crashes, TK, or "grab", Lexine as she bends over to pick up her helmet. Or really, for you the viewer.
Flushed 
Defeat the giant necromorph in the water tunnels
The first boss in the game is at the end of chapter 6.
The first stage is to shoot the yellow points, and then when you come to a fan, use stasis. The fan will kick back up and cut the tentacle. Repeat on right tentacle.
Stage 2 requires you to TK the explosive barrels, and fire them at the tentacles as they approach.
Stage 3 has you TK the explosive goo he fires at you and throw it back at the small yellow orb at water level or use the arc welding gun. (this works really well).
Lexine, I Told You To Be Quiet!
Defeat the Brute
The brute reside in chapter 7, and really only has two stages.
Stage 1 will have you stasis the brute when it is close and charging. Shoot the yellow on its back to damage it. TK any acid it throws. Eventually it will rear back. Shoot its should to rip off an arm.
Stage 2 is much quicker, and all you have to do is throw back the acid it will be spewing at you.
Sack Blast
Defeat the giant necromorph on the hull of the Ishimura
This boss is found at the end of chapter 9, and will cycle through multiple stages as you go.
Stage 1: I would advise using the Pulse Rifle for this stage. Shoot the yellow bulbs as on the tentacles as you can see them. after three or four times stopping the tentacles, this stage will end.
Stage 2: I would advise using either the Flame Thrower or Rivet Gun. This is primarily to save ammo for the Pulse Rifle. In this stage, the boss will try to eat you. Small yellow pustules will surface. If you don't shoot them quickly, they will fire acid blobs at you. Quickly shoot each one, and the monster will pull away. Shoot the yellow mouth a couple times and this stage is complete.
Stage 3: I would advise using the Line Gun for this stage. A string of orange balls from its mouth. Shoot it nearest to the mouth as you can. This will destroy all the orbs at once, and save you ammo. Several leapers may drop in this stage, but that isn't a bad thing. Killing them will often drop health.
Stage 4: I would advise using either the Pulse Rifle or Contact Beam. Quickly shoot the small orb that appears at the of the appendage. If you use the Contact Beam, just fire a charged shot at it.
Now you just need to repeat that four times. If you mess up stage four at any point, you'll basically have to repeat all the stages for that round. This boss will go down after completing stage 4 five times.