Some chronological information has been given throughout Discovery, but much of it was vague. However there were some concrete dates given, and enough vague information to form a reasonably specific framework for when each episode takes place. I will examine each episode in turn, and slowly build a framework for the series as specific as possible.
I originally undertook this project with the help of the other Star Trek Timeliners. Thanks to them for the inspiration and aid, and letting me post all this here. I developed my own interpretation of these dates, based on including reference to Klingon time periods, so this is somewhat closely related to the other work I've done here on Klingon calendar systems.
It should be noted that the Discovery stardates jump all over the place and are thus just as useless for chronological purposes as TOS stardates, perhaps moreso, as I know there are some who would accept a stardate based approach to TOS chronology, but that would be impossible with Discovery.
Also of note is that I haven't (and may not) watched past season two. So the following breakdown only covers those first two seasons.
1x01-1x02- "The Vulcan Hello" and "Battle at the Binary Stars"
It's specifically stated that the episode starts on May 11, 2256. The final scenes of the episode take place some time later, but there's no clue as to how long. We'll just assume that it's not any farther in time than the end of the next month, so June 2256.
1x01-1x02- May 11, 2256
1x02 Epilogue- Range of May 12- June 30, 2256
1x03- "Context is For Kings"
Six months have past since the end of the last episode. Allowing "six months" to mean anything from five and a half to six and a half months, and continuing with the idea that the last episode ended sometime between May 12, 2256 and June 30, 2256 we end up with a range of October 27, 2256 to January 15, 2257 for the first day of this episode. It spans four days.
1x01-1x02- May 11, 2256
1x02 Epilogue- Range of May 12- June 30, 2256
1x03- Range of October 27, 2256- January 15, 2257
1x04- "The Butcher's Knife Cares Not For the Lamb's Cry"
One day after the end of the previous episode. Since that took place over four days, this gives us a range for this episode of October 31 to January 19.
It is also said, in Klingonese, that it has been six months since the Battle of the Binary Stars. Usually in Star Trek, we could assume that dates spoken by aliens have been translated to Earth time periods. However, as Discovery shows Klingons speaking in Klingonese with subtitles, I prefer to conclude that this actually refers to six Klingon jar. According to the Klingon calendar system used many times in the novels (see here), each Klingon jar takes 40.57 Earth days. So again assuming this refers to something between five and a half and six and a half jar, we can arrive at a range of dates this episode could fall into.
May 11, 2256 converts to 47-48 A'Kahless, Year of Kahless 881 in the Klingon calendar system. Five and a half Klingon jar later would be 25 Doqath, Year of Kahless 882. Six and half Klingon jar later would be 24 Xan'lahr, Year of Kahless 882. These dates convert to December 22, 2256 and January 30, 2257.
As will prove even more important later, however, the six month ago reference could also refer to six calendar months having past since the event. Just as July 1 could be said to be six months after January 31, since the calendar pages have flipped six times in that span, a date earlier than five and a half months after the Battle of the Binary Stars could conceivably be referred to as six months later. In this case, as the battle took place on the final day of the month of A'Kahless, any time in the month of Doqath could be referred to as six months later. 1 Doqath, Year of Kahless 882 converts to December 2, 2256. This gives us a more acceptable start date for the range this episode can fall into, as will be seen later.
Accounting for the latest date the previous episode could fall on, we can trim the end date of the range back and arrive at somewhere between December 2, 2256 and January 19, 2257. And since the previous episode has to fall four days before this one, we can trim the start date of that range as well. Similarly, if 1x03 was six months after the epilogue scene of 1x02, we can trim the beginning date of that range back.
1x01-1x02- May 11, 2256
1x02 Epilogue- Range of May 16- June 30, 2256
1x03- Range of November 28, 2256- January 15, 2257
1x04- Range of December 2, 2256- January 19, 2257
1x05- "Choose Your Pain"
It's been less than three weeks since the last episode. I could interpret this to be anywhere from eighteen to twenty days later. This gives us a new range of December 20, 2256 to February 8, 2257 for the first day of this episode. Though the episode is not perfectly clear, I believe it takes place over two days.
This episode is also said to be seven months after the Battle of the Binary Stars. Stretching that to seven and a half months, this episode couldn't take place any later than December 26th. This reference shows why the earlier date was necessary for the range I developed by interpreting the Klingon dates as I did. So we know we are on the right track, and there is nothing inconsistent with using the Treklit version of the Klingon dating system.
Tightening up the farthest end date for this range forces us to also tighten the end date for the previous episode's range, as it can't take place less than 18 days previous. This again trims the range for 1x03 and 1x02.
1x01-1x02- May 11, 2256
1x02 Epilogue- Range of May 16- June 19, 2256
1x03- Range of November 28- December 4, 2256
1x04- Range of December 2- December 8, 2256
1x05- Range of December 20- December 26, 2256
1x06- "Lethe"
Less than a week has past since the last episode. I interpret this as between four and six days. So assuming the last episode took place over two days, this episode occurs some time between December 24 and January 1. I'd say this episode also takes place over two days.
1x01-1x02- May 11, 2256
1x02 Epilogue- Range of May 16- June 19, 2256
1x03- Range of November 28- December 4, 2256
1x04- Range of December 2- December 8, 2256
1x05- Range of December 20- December 26, 2256
1x06- Range of December 25, 2256- January 2, 2257
1x07- "Magic to Make the Sanest Man Go Mad"
No real chronological clues in this episode, other than a possibility that the party depicted in the episode could be a New Years Eve party. Though there are no overt signs of it. It does seem that a week or more has past since the last episode.
1x08- "Si Vis Pacem, Para Bellum"
No clue how long it's been since the last episode, but this episode seems to span two days.
1x09- "Into the Forrest I Go"
Picks up the same day as the last episode.
It is said that Tyler's imprisonment was exactly 227 days long. This could either mean that there were 227 days between the first day of his captivity and the last, or that he was freed on the 227th day, meaning that there were 226 days between the first and last day. This span is between episodes 1x01 and the second day of 1x05. Since we know 1x01 took place on May 11 that means the second day of 1x05 occurs on either December 23 or 24, 2256. Amazingly this falls within the very specific range that I have developed during this process for that episode.
We can tighten up all many of the other ranges based on this more specific date.
1x01-1x02- May 11, 2256
1x02 Epilogue- Range of May 16- June 17, 2256
1x03- Range of November 28- December 1, 2256
1x04- Range of December 2- December 5, 2256
1x05- Range of December 22- December 23, 2256
1x06- Range of December 27- December 30, 2256
1x07- Range of December 28, 2256- January 2257
1x08- 2257
1x09- 2257
1x10- "Despite Yourself"
Picks up the same day as the last episode. No other clues are given, but the episode seems to take a few days.
1x11- "The Wolf Inside"
The day after the end of the previous episode.
1x12- "Vaulting Ambition"
The same day as the "The Wolf Inside." Still no clue as to when relative to the first have of the season this arc takes place.
1x13- "What's Past is Prologue"
The day after "Vaulting Ambition." But the episode concludes nine months after "Into the Forrest I Go." 1x08-1x13 seem to only take place over about a week's time. Given that there is an indefinite gap between 1x07 and 1x08, we will have trouble nailing down exactly when this nine months gap ends. If the time between 1x07 and 1x08 is short, this episode could end in October. If it is anything more than 3 months (which seems too long), this episode ends sometime in 2258.
1x01-1x02- May 11, 2256
1x02 Epilogue- Range of May 16- June 17, 2256
1x03- Range of November 28- December 1, 2256
1x04- Range of December 2- December 5, 2256
1x05- Range of December 22- December 23, 2256
1x06- Range of December 27- December 30, 2256
1x07- Range of December 28, 2256 - January 2257
1x08- 2257
1x09- 2257
1x10- 2257
1x11- 2257
1x12- 2257
1x13- 2257
1x13 Epilgoue- Range of October 2257- 2258
1x14- "The War Without, the War Within"
No real clues, other than that it picks up directly after the previous episode.
1x15- "Will You Take My Hand"
Again no concrete clues, but one thing of note. Burnham says the first episode occured "a year ago", so while quite vague (we know it was at least seventeen months ago), we can assume that means this episode is still in calendar year 2257, and so referring to May 11, 2256 as "a year ago" is accurate. This makes the rest of the undated 2257 episodes take place no later than March, to still be nine months before.
The final scenes take place an unstated amount of time after the rest of the episode. Future information, both in Season Three, and in the novel Die Standing imply that by the final scenes, we have crossed over into 2258. This also gives a clean break between years for each season.
1x01-1x02- May 11, 2256
1x02 Epilogue- Range of May 16- June 17, 2256
1x03- Range of November 28- December 1, 2256
1x04- Range of December 2- December 5, 2256
1x05- Range of December 22- December 23, 2256
1x06- Range of December 27- December 30, 2256
1x07- Range of December 28, 2256 - January 2257
1x08- Range of January - March 2257
1x09- Range of January - March 2257
1x10- Range of January - March 2257
1x11- Range of January - March 2257
1x12- Range of January - March 2257
1x13- Range of January - March 2257
1x13 Epilogue- Range of October - December 2257
1x14- Range of October - December 2257
1x15- Range of October - December 2257
1x15 Epilgoue- Early 2258
"Section 31 Bonus Scene"
No clues at all, other than that it takes place after Season One.
Short Treks "Runaway"
No clues at all, other than that it takes place after Season One.
Short Treks "Calypso"
No specific date other than "almost a thousand years" after "the crew left." Given the events of the second season finale, at this time I am left to conclude that this takes place at least around the year 4000.
Short Treks "The Brightest Star"
No specific dates other given. Interviews with the writers point to about 2238. 2x06 dated these events as "18 years ago", so assuming 2x06 takes place in 2257 that makes this short 2239.
Short Treks "The Escape Artist"
While much could be said about the implied timeframe of this short, weather pre-I, Mudd or post- I, Mudd, in the end there is no concrete chronological information given, other than that it must take place at some point after Mudd's previous Discovery appearances.
2x01- "Brother"
Picks up directly after the epilogue of 1x15. Seems to take place over the course of two days.
2x02- "New Eden"
Picks up directly after the last episode. Takes place over two days.
2x03- "Point of Light"
No indication of how long since the last episode, but this one takes place over at least two days.
2x04- "An Obol for Charon"
Takes one day. New Eden took place "this week". And Brother was last week. This means there's probably no gap between 2x02 and 2x03, or 2x03 and 2x04.
2x05- "Saints of Imperfection"
I think this episode takes two days. It seems to pick up on the same day as 2x04 and continues until the following day.
2x06- "The Sound of Thunder"
I would say this has to be the same day as 2x05, and only the following day from 2x04. Culber's return is very recent, and Saru is getting his first medical exam in the wake of 2x04. I would guess this episode spans at least four days, depending on travel time.
2x07- "Light and Shadows"
The Burnham portion of this episode seems to take several days, while the Pike portion takes only five hours. This discrepancy is probably accounted for during the next episode, where everything syncs back up. It's hard to pinpoint a day count for this though.
2x08- "If Memory Serves"
I'm just going to guess that by the time this episode is over, almost a week has past since 2x06. So little hard evidence to go on.
2x09- "Project Daedalus"
Takes place during one day. Seems to begin the day after 2x08.
2x10- "The Red Angel"
Picks up on the same day as 2x09. I think it spans at least three days.
2x11- "Perpetual Infinity"
Stars mere hours after 2x10 and only takes up two hours.
2x12- "Through the Valley of Shadows"
It seems this episode starts the day after 2x11, and proceeds at least into the next day.
It's said it has been months since the events of 2x03. I can only concretely account for 17 days having elapsed. Therefore there must be some significant travel time not accounted for in many of these episodes, specifically 2x06, 2x07, 2x08, and 2x10. If an extra week of travel time took place during each of these episodes, then we would be approaching a time period that could reasonably be called "months".
2x13- "Such Sweet Sorrow, Part 1"
Picks up just after 2x12 and takes place on that day, or through the next day at the most.
2x14- "Such Sweet Sorrow, Part 2"
Picks up the same day as 2x13. The final scene takes place 124 days later.
Based on evidence in Season Three, and the novel Die Standing, this episode takes place in 2258. Serendipitously, there is just enough time after the Enterprise's refit for Pike to command another five year mission ending in 2263 and then pass command off to Kirk.
Short Treks "Q&A"
There is no chronological data in this episode. Never the less, everything we know about Spock's career, from canonical and non-canonical sources, leads me to place this event in 2253.
Short Treks "The Trouble with Edward"
No chronological data is given in the episode, but we can assume this takes place after Discovery Season Two, sometime in 2258.
Short Treks "Ask Not"
Again no chronological info is given, but we can assume this takes place after Discovery Season Two, sometime in 2258.
So to simplify those ranges of dates, we are most likely left with this vaguer but more useful information: