𝐖𝐄𝐋𝐂𝐎𝐌𝐄!
There is little continuity within the heists in the first game, with only the order of the heists in the heist selection menu indicating some sort of timeline.

All that is known for certain is that First World Bank is the first heist in the game, as based on its position on the list in the heist selection screen, the description of the heist, newspapers in Counterfeit, and the description of the heist's PAYDAY 2 remake plus comments by Bain, and Counterfeit occurs after at least Slaughterhouse, if not every heist in the game, as when putting the drill on the Frank Yaeger safe, Bain will comment on how unlike the "other ones", this drill won't jam. It is also presumed Undercover is the last heist to take place in PAYDAY: The Heist due to it being at the bottom of the heist selection screen list.

It has been confirmed by both Overkill Software and Valve Corporation that No Mercy is not canon in either the PAYDAY universe or the Left 4 Dead universe. However, this changed in PAYDAY 2 in which the stolen virus ended up in the hands of Murkywater and is used to infect Bain after he was captured by them.

Hoxton Arrest (2012)
At an unknown time between the setting of the first and second game, the original Hoxton was left behind after a job gone sour. However, all of this was after the first PAYDAY game as his lawyer in "Intro" song in the Merry Payday Christmas Soundtrack says "Look, they got you. You understand? They got you good. You were caught on tape at the Garnet Group break-in, your fingerprints were all over the panic room, and your apartment is filled with blueprints from banks in D.C." This suggests that they had already moved to D.C. He was arrested by the Federal Bureau of Intervention and incarcerated at the same high security prison, Hazelton, and cell as their previous driver, Matt, whom he regularly beats up, which he sees as "karma" for double-crossing them.

PAYDAY 2 (2013-Present)
It has been confirmed by Overkill that PAYDAY 2 takes place after PAYDAY: The Heist.

In the tutorial heists, Bain mentions it has been two years since Dallas has last participated in a heist, which would align with the release dates of PAYDAY: The Heist and PAYDAY 2.

There were also various boxes in the pre-Update #110 safe house named after various heists from the first game. It is speculated that these boxes contain loot or equipment from those heists.

Pre-FBI Files Release
Until the FBI Files were released, confirming PAYDAY 2's timeline, the order in which the heists took place was almost entirely up in the air. All that was known at the time was that Hector's heists took place in a definite order, along with The Dentist's and The Butcher's heists, though the order of The Dentist's heists are less defined. Additionally, some voice lines indicate a sort of order regarding Vlad's heists.

Hector Timeline
It was known for certain that Watchdogs takes place before Firestarter and Firestarter before Rats, both based on Hector's comments at the end of the former, and the elimination of the Mendoza cartel's D.C. operation in Rats would make Firestarter occurring after Rats impossible.

After the PAYDAY crew fulfills his contracts, Hector tries to seek protection from the FBI but is eventually assassinated in Hoxton's Revenge. His death would be mentioned later in the Goat Simulator heist in which a "void in the power trade" occurred without him in power.

The Dentist Timeline
Even though not one of The Dentist's heists, he and his heists put The Elephant's Election Day heist along The Dentist's timeline.

In The Dentist's introductory video, he mentions the First World Bank and Election Day heists while talking to Dallas, along with teasing some of his heists, supporting that Election Day occurs before any of The Dentist's heists.

Hotline Miami is The Dentist's second heist, following The Big Bank. In the heist, the Payday Gang kills The Commissar so The Dentist can convince the District Attorney, whom has relations to the Commissar, to lower security around Hoxton. Since the Election Day heist is committed so The Elephant can have Hoxton be transferred to a lighter-security prison, Hotline Miami would have to occur after Election Day, as if it were the other way around, the act of lowering the prison's security would be moot, since Hoxton would be transferred to a different prison anyway.

The Dentist's next two contracts are Hoxton Breakout and The Diamond, though it was unclear in which order. In the Hoxton Revenge heist, The Dentist's fifth heist, evidence boards show footage of Clover breaking into the FBI office with the rest of the crew in Hoxton Breakout, even though she was released alongside The Diamond, released after Hoxton Breakout.

The Dentist's last heist is Golden Grin Casino, taking place after each of his previous heists. Said heist takes place before the Reservoir Dogs Heist, Breakin' Feds, and Henry's Rock with Bain's capture and the box bearing very similar markings like the other ones taken by the crew. In Hell's Island, it is revealed he has turned into a traitor and has joined forces with Murkywater.

He is eventually killed by the crew in The White House for holding Bain at gunpoint in an attempt to stop them, should the crew solve the secret.

The Butcher Timeline
The Butcher's two heists occur in a "What if?" scenario.

The Butcher's Forest contract takes place if the crew doesn't do the Dockyard contract, and the bomb is loaded onto the train that the gang assaults in the Forest heist. However, neither heist retcons the other, both heists canonically taking place in the PAYDAY universe.

After The Bomb: Forest or The Bomb: Dockyard, The Sosa cartel sinks one of her ships, then contracts the PAYDAY gang to kill Ernesto Sosa.

While not the contractor, The Butcher's men are involved in Alaskan Deal.

Vlad Timeline
In Meltdown, which takes place after Shadow Raid as indicated by the open, empty samurai armor vault, when one of the warhead cases is opened, Bain may reference the Mallcrasher and Ukrainian Job heists ("Vlad, smashing malls is one thing, stealing tiaras is one thing, but nukes?!"), indicating that those two heists take place before Meltdown, and likely Shadow Raid.

In Aftershock, when one of the safe trucks is opened, Bain will reference the Ukrainian Job and Meltdown heists, further reinforcing the heist order in the files.

Goat Simulator takes place sometime after Hector's death (and therefore, Hoxton's Revenge), in which Vlad mentions the death of the former drug lord created a void in the powder trade.

Post-FBI Files Timeline
The release of the FBI Files not only confirmed that the heists added to the game post-release take place in the order they were released in, clearing up the mystery around Hoxton Breakout and The Diamond, it also confirmed the order the heists in the game since its release are in. It is shown in the Heist Database that Four Stores is not only Vlad's first heist, it is also the first heist in PAYDAY 2. It also confirms Jewelry Store to be Bain's first contract and Big Oil to be the Elephant's first. It also shows that Beneath the Mountain is Vernon Locke's first contract, which is also supported by Bain voicing suspicion upon completion of the heist

Jimmy Timeline
Murky Station takes place before Boiling Point, as in Murky Station the crew and Jimmy steal the EMP bomb they need to access the hidden lab in Russia in Boiling Point. Also, Murky Station occurs after Locke's Beneath the Mountain and/or Birth of Sky heists (but allegedly, prior to Alaskan Deal), as Bain mentions stealing the EMP bomb would earn favors for him.

Vernon Locke Timeline
At some point after the PAYDAY crew conducted heists on Murkywater such as Shadow Raid and Meltdown, Vernon Locke man