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Complete all Trials.

Online required. For this, you have to complete all the single player and multiplayer trials in the base game. Thankfully, everything is cumulative (i.e. it's not like Dead Rising 3 where when you finish tier 1 of a trial, tier 2 starts from 0 again).

For more info on the single player trials, refer to "Story Completionist." As for the multiplayer trials, they are as follows. If I didn't list what a particular rank unlocks, it's because it just unlocks PP and/or money:

Create 1, 5, & 10 gold combo weapons
Self-explanatory. Obviously you have to unlock gold weapons to be able to make them. You can find gold bars as random drops, or buy them from vending machines once you've done the "loot 20000 scrap" trial. The last rank unlocks the Solid Gold skill.

Kill 1, 10, 25, & 50 bosses
As noted in the overview, these only spawn on Day 2 of an episode. After you complete or fail all the missions given to you at the start of a day, all players must go to Kiichiro Plaza to ring the Doom Gong. It's recommended to farm Episode 1 or 2 for this, as they're relatively easy. Regardless, this is a real grind - it is likely the only multiplayer trial that will take longer for you will be the 250 missions one, and it's very possible this one will actually take more time.

Rank 1 unlocks the Spray & Pray combo weapon, which can be used to heal allies. Rank 3 unlocks the Gold Electric Axe, and 4 unlocks the Gold Ice Sword.

Kill 1, 10, 25, & 50 loot zombies
Loot zombies are fresh zombies that are dressed like snowmen or gingerbread men and are wearing backpacks. This will come naturally as you play. Ranks 1-3 unlock tiers of the Grave Robber skill; rank 4 unlocks the Gold Equalizer.

Complete 1, 25, 100, & 250 missions
Likely the last multiplayer trial you'll finish, this is the biggest grind of the multiplayer trials - particularly because the "kill X number of zombies" trials don't count, unless they're given to you at the start of the day. Once you've leveled up a bit and taken care of the other trials, I'd recommend playing the later episodes - even though they're harder, they have more missions you can complete.

Rank 3 unlocks the Gold Blambow; rank 4 unlocks the Gold Ornament Gun.

Complete any 3 missions in episode 1
Unlocks episode 2. They do not have to be done in one game.

Complete any 8 missions in episode 2
Unlocks episode 3. They do not have to be done in one game.

Complete any 12 missions in episode 3
Unlocks episode 4. They do not have to be done in one game.

Complete any 15 missions in episode 4
Unlocks exo suits. They do not have to be done in one game.

Complete 1, 10, 25, & 50 days
"Completing" a day means surviving it, i.e. getting to the safehouse at the end. Ranks 1-3 unlock tiers 1-3 of the Quick Study skill; rank 4 unlocks the Gold Bazooka Cannon.

Revive another player 1, 10, 25, & 50 times
This is best done via boosting, as it's a real grind otherwise. Load up an episode 1 private match with a friend, and proceed to one of the nearby escalators outside of Medieval Town. Take turns throwing yourselves off the 2nd floor and reviving one another. If one of you is leveled up a bit already, it's recommended that person finish his or her 50 revives first. They should then start up a new game in a different save slot - their character will start at level 1 again, meaning they'll go down super quickly. Once both people have the revives, the person who loaded a new game can go back to their original one.

Rank 2 unlocks the Thick Skin skill; rank 4 unlocks the Gold Laser Sword.

Heal others for 1000, 5000, 10000, & 20000 health
This is relatively easy enough, even without boosting. You can just follow another player throughout the episode and heal them as necessary. The hardest part of this is actually getting the healing items - you'll either have to play as Connor (he starts with one), unlock the blueprints for them so you can make them, buy them from vending machines, or find them as random drops.

Rank 4 unlocks the Rocket Bandage combo weapon.

Loot 5000, 20000, 50000, & 100000 scrap
Self-explanatory. These will come naturally. Rank 2 unlocks the Investment skill; rank 4 unlocks the Jackpot skill.

Eat 10, 20, & 30 times in one day
Can be done solo. Basically keep grabbing a bunch of food, run into crowds of zombies, and heal yourself every time you get hit. Best done on episode 1. Rank 1 unlocks the Potluck skill.

Survive a day without anyone dying or eating food
Unlocks the Food Bank skill. Can be done in solo private. It's easier the higher your level. If you're having trouble, you can do this in day 2, and just proceed to the safe house (see the overview for locations) and wait there.

Survive a day and complete a mission without anyone killing a zombie
Unlocks the Frostbite skill. Can be done in solo private. Obviously some luck is required, as the mission itself has to be one that you can complete without killing anything. I'd recommend hanging out at the safehouse location/finding somewhere you won't be attacked after you complete the mission.

Finish a day where everyone is revived once, but no one dies ("I Get Knocked Down")
Unlocks the Surgeon skill. You'll likely have to boost this in a private match (it can be done with 2 people), as this doesn't happen naturally very often (or ever). Everyone is either good enough to not go down at all, or bad enough that they die quickly.

Kill the most zombies in a day 1, 5, 10, & 15 times (Zombie Killer MVP)
Requires playing with at least 1 other person. Rank 3 unlocks the Out With a Bang skill, rank 4 unlocks the Gold Ion Cannon.

If you're trying for this legit and are having difficulty, I have two words for you: Split Shot. Either buy 2-3 of them or make them (the latter requires having the blueprint from single player, and then combining a lead pipe with any regular firearm). It's also preferable to have bought the skills that increase your ammunition for assault rifles (the Split Shot maxes at 480 bullets, which means you can fire continuously for a minute or two without stopping). After that, go to Caribbean Cove where the pirate ship is, and unload on the zombies there - there are usually several hundred in the area. Other good locations are the mall entrance (southeast part of Central Plaza) and the lower floor of the Amazon Food Court, though the latter depends on what episode you're on.

Kill the first zombie in a day 1, 5, 10, & 15 times
Requires playing with at least 1 other person. If you're having trouble with this, you can shoot zombies through the fence at the start of the day. Just be considerate of others and don't do that once you've unlocked the max rank for this.

Rank 2 unlocks the Gold Floating Lantern, rank 4 unlocks the Gold Acid Maul.

Reach the safehouse with less than 50 health.
Can be done solo. Find some zombies when you're near the safehouse at the end and get hit until you're below 50 health. It could be useful to have a partner to revive you in case you lose too much health.

Lose more than half your health from falling damage without dying
Easiest done when you're a low level, as once you level up the percentage of health you lose when you fall will be less. Just find somewhere that has a 2nd floor and jump off.

If you do goof up and level up a lot before trying for this, you'll have to lower your health significantly before jumping; you have to lose more than 50% of your current health (not your max health).

Heal 3 different people in a single day
This is in a single game day, not a real-time day. Obviously requires playing with a full game of 4 people. You'll need to either unlock healing items by completing multiplayer trials, find those items as random drops, buy them from vending machines, or play as Connor, since he starts with the healing spray weapon. Note that it only has 60 ammo, so be careful with it.

Kill 500 zombies in a day as a team
Can be done solo, but is obviously easier as a team. See the method above for Zombie Killer MVP if you're having trouble.

Be the first to arrive at the safe house 1, 10, 25, & 50 times.
This is the most troublesome of the online trials that can't be done solo, since it's a grind, and you can only get one per day. As such, this should be priority number one if playing legit. As noted in the overview, keep an eye out for stores with flashing red lights, as one of these will be the safe house in day 1; as for the day 2 safe houses, refer to the overview for their locations.

The first 3 ranks unlock the various tiers of the Stock Up Skill; the last unlocks the Gold Flaming Sword.