- Story related, cannot be missed -
Bare information: you will get this after successfully solving Celso's problem in Rubacava.
Missable Trophy Checklist (in a chronological order, of sorts):
* If you've missed any of these, I'd simply suggest creating a new game to go ahead and claim them.
*EDIT: Until the trophy data is updated, the above trophy is called “It’s like I’m not happy unless I’m breathing in the thick, black, nauseating fumes…”
*** Spoiler - click to reveal ***Immediately upon starting out, head towards the front of your office, without heading out. You should change the camera angle to where you can look at the table in your office. Pick up the cards here and then you can go ahead and leave. Walk up to Eva and inquire into her personal life before her death, and then walk around to the front of her desk and ask to use her hole puncher. Take the first elevator down the hall and you'll see Domino speeding off. Head further into the garage and you will encounter Glottis. Before you inquire about the car and why yours isn't available, ask him about his name and more about him. After that, feel free to ask about the car and any modifications Glottis could do to it in exchange for becoming your driver.
Take the work order back to Eva and you will be shot down. However, something seems fishy about the response. Take the second elevator and head outside, going straight down the hall on the first floor. Outside, turn right and go off-screen from there. A parade can be seen off in the distance, so head towards it. You will find a sad looking skeleton clown and a bread stand. Ask the clown for a Robert Frost balloon and two worm balloons. Pick up the bread nearby, and then head off to the alleyway near the Department of Death building. In the back, you can see a rope leading up. Climb up it and enter the first set of windows you come across. Aha! So your boss likes to play hookey! Investigate the nearby computer to find out how to change the response to your favor. Return to Eva, get the slip signed, and when you regain control, you'll be in the Land of the Living.
Walk over to one of the hodge-podge living humans and talk to them three times, having fun scaring them. Then walk over to the rattling burrito-like feature and press 
, immediately getting your scythe out. Open the package up and, when you regain control again, head back down the elevator to encounter Brennis, the mail demon. After he stomps off, go back to the Packing Room where you just saw the poor soul enter and take out your worm balloons (basically deflated balloons). Use one on the right spout, then use the other on the left spout. The mixture hardens like packing foam, but we don't want the mixture itself in our pockets. Now, head back to Manny's office, over to the message tube, and put both balloons down it. You'll see a video of the sorting machine stopping. Head back out and, before taking the elevator back down to the first floor, stop by Eva's desk once again and stand before the hole puncher. Take out the deck of cards and use it on the hole puncher.
Back on the first floor, go to the opposite side of the lobby to find the Sorting Room. Brennis will be working on the machine, annoyed at the state it is in. You can pick up the fire extinguisher now, or wait and talk to Brennis about everything. Once you've exhausted all topics, a small fire will spark on his clothing, which will signal the requirement of the extinguisher itself. At the large bolted door, play with the deadbolt, locking it. After that, leave and come back. Brennis will be gone and, since you locked the door, it wouldn't have closed all the way. Open the sorting room then head over to the red tube. Take out the punched card and use it on the tube, getting yourself the info on a new client.
With Mercedes now safely back in your office, ask her about any dirty deeds that would've stopped her from entitlements towards the Ninth Underworld. After the disappointment, head down the hallway to be called in to your boss's office. When you regain control, try the door and insist on taking down the corruption in the DoD. Speak to Salvador when you are able to move once again and ask him about everything he knows thus far, learning of the shadows that linger in El Marrow. On the other side of the room you'll find Eva, the secretary, who is really a spy. Talk to her and ask her if there are any new messages for you, then find out what both of them want. Salvador wants some pigeon eggs and Eva wants your teeth.
Head back outside and then climb up the rope. Your "boss" will be in a meeting, but if you go around the left side of the screen, the camera will shift and you can, instead, enter Domino's office. Enter it and try out his computer a few times, then go further in and take your anger out on the punching bag three times. Pick up the mouth piece, then investigate Domino's desk drawers. That was all you could get out of his office. Leave back onto the windowsill and head towards the rope but don't head down. Instead, look onto the other side and collect the remaining rope. Dismiss the item to have it pool on the sill beside you, then take out the glowing coral you got from Domino's office. Use it in conjunction with the rope to get access to the roof, where the pigeons are located. Head near them so the camera shifts and stand beside the feeding dish. Take out the Robert Frost balloon and turn to face them, pressing 
to activate it. He'll try to scare them, but the pigeons won't understand. Instead, put the balloon in the feeding dish, then take out the bread and use it on the dish as well. Once they are scared off, go collect the eggs and head back down to the ground floor.
Before you return to the LSA, re-enter the garage through the new opening and head back to where you were incarcerated. Take out the mouthpiece and use it on the coffee machine in Glottis's little shack. Then turn around slightly to where Manny isn't looking at anything in particular and use the gooey mouthpiece, casting a mold of your teeth. Take both of these items back to Salvador and you will be taken outside El Marrow, to the Petrified Forest. Off-screen, you will find Glottis, heartbroken. He'll demonstrate how they ripped his heart out, so it is up to you to fix it. Head behind Glottis to the demon spider's den and walk over to the web. Pick up a few bones (you only need one for this part) and throw one into the web. With it firmly placed, take out your scythe and use it on the bone, jettisoning the heart back near Glottis. Return, pick it up, and give it back to your good ol' buddy.
Now how are you going to get out? Easy, with the Bonewagon! Driving the thing, head to the middle right exit and you should come to a pump near a strange tree. When you get out, Glottis will reappear pushing a wheelbarrow. Talk to him about what it's doing, then pose the wheelbarrow until the tree's pumps start to shake the tree up by being off-balance. With this sign, turn the machine off and Glottis will attempt to work on the balance some more. While he's up there, turn the machine back on and try to contain your laughter. Now the Bonewagon can traverse up rocky slopes, but we are not quite done. Head back to the first screen where you found Glottis and get out. Pick up the sign and then go off the left side, coming out into a large opening. Use the sign in a game of hot-or-cold, watching where the arrow points to and taking it further in that direction. Eventually, the arrow won't spin or point, and you'll have hit the sweet spot. Return to the Bonewagon and drive down the new path, listening to Glottis complain about the sign. From this sign, go pick up the key and then return to the wagon, driving up the nearby rocky path.
All that is left is taking the top right passage on the first screen of the forest. Drive up it to encounter a large gate. Glottis knows what's behind it and tells you directly, but you have to go through it. When you enter, you'll see a large dam of bones and demon beavers. You cannot take them on directly, even with the fire extinguisher. They'll just relight on the nearest beaver, and while you can eliminate three of them, the fourth is impervious. So, you have to be crafty. Near the bottom left of the path, you can find a deviation leading to a small spot under a rock. The beavers can smell you but can't see you, so hide here. If you still have a spare bone or two, use one and toss it into the tar. Immediately take out your fire extinguisher and aim it directly where the bone was tossed, holding the button to spray it. The beaver will hiss and jump towards it, going right through the stream and becoming doused before impact. This will stick them in the goo and render them the same way some dinosaurs died. Repeat this to clear the way towards Rubacava. If you run out of bones, head back to the dam and press X near the edge between bone and dirt, as you can pick up some spares there. Once all of the beavers are gone, return to the front of the gate and use the key to open it.
Welcome to Rubacava! Head up the stairs and off into the fog to be rescued by Velasco. Now, head up the stairs further and enter the automat to speak to Celso. Ask him about getting a job there, and after all that, ask him why he's still here. Return to the Bonewagon, where Velasco will be hanging out, and ask him about the eyepatch, then about Celso's beloved. With the logbook in hand, return to Celso and show him to bad news. Congratulations! You've successfully landed yourself a new job.
If any of the information I've provided is incorrect, please let me know so I can change whatever's necessary to reflect it.