(from his posting February 2012)
Now, not all of the dates listed above are what you might call canonical. Some of them are, but some are just wild-ass guesses on my part. Here's where they all came from:
"A Boy’s Best Friend": Given the speed with which space is being explored and settled, a date of 1980 for the establishment of Lunar City seems reasonable. Assuming Anderson Senior was one of the original settlers, that places the story in 1995.
"Robbie": Stated in I, Robot.
"Robot AL-76 Goes Astray": After robots are banned from Earth in 2003-2007, before Susan Calvin joins US Robots in 2008.
"Insert Knob A in Hole B": Before the use of robots on space stations, hence before "Reason".
I Robot: To Protect: As stated in the novel, 2035. Based on Susan Calvin's age, given her birth in 1982, the story should take place in 2008.
I Robot: To Obey: As stated in the novel, 2036. Based on Susan Calvin's age, given her birth in 1982, the story should take place in 2009.
"Runaround": Stated in I, Robot.
"Reason": Six months after "Runaround".
"Catch That Rabbit": Six months after "Reason".
"Liar!": Stated in I, Robot.
"Satisfaction Guaranteed": As stated in the story, takes place fifty years after World War II, i.e. in 1995. However, the characters have the same titles as in "Liar!", and Susan Calvin is more knowledgable about emotions, so the story takes place after "Liar!". Also, Tony has more advanced vision than Dave in "Catch That Rabbit!"
"Balance": Susan Calvin’s robotic servants flatter her in a manner similar to Herbie from "Liar!", but she seems much more at ease with the idea, which places the story after "Liar!"
"Blot": Story of the first exploratory mission to Miranda. Given a mission to Mars in 1998, a first expedition to Mercury in 2005, and bases on Titan in 2025, a mission to Uranus in 2026 seems reasonable.
"Little Lost Robot": Stated in I, Robot.
"Cal": Cal seems to be the prototype for the EZ robots of "Galley Slave", so allowing a few years for the design and production of the latter places the story in 2031.
"Lenny": Peter Bogart is now Senior Mathematician, so the story comes between "Little Lost Robot" and "Risk".
"Evidence": Stated in I, Robot.
"PAPPI": Immediately after "Evidence".
"Risk": Takes place ‘some years’ after "Little Lost Robot".
"Escape": Within a few months of "Risk". (In I, Robot, "Escape" comes immediately after "Little Lost Robot", but logically, ought to come after "Risk".)
"Galley Slave": Stated in the story.
"First Law": Stated in the story.
"Plato’s Cave": Shortly before Stephen Byerly becomes Regional Coordinator.
"The Evitable Conflict": Stated in I, Robot.
"Robot Dreams": Late in Susan Calvin’s career with US Robots.
I, Robot: The links between the stories are set fifty years after Susan Calvin joins US Robots in 2008.
"Feminine Intuition": Five years after Susan Calvin retires from US Robots in 2058.
"The Fourth Law of Robotics": Susan Calvin’s great-niece has her job.
"Christmas Without Rodney": Slighting reference to the 20th century suggests the story takes place in the 21st. Common use of robot servants suggests the latter part of the century, when the Frankenstein Complex has faded away.
"Kid Brother": Acceptance of household robots on Earth dates the story to the same general era as "Christmas Without Rodney" and "Light Verse".
Robots in Time: Stated in volume 1, Predator.
"Light Verse": Features a robot that is capable of original artistic expression (as was Andrew Martin), at a time when robot servants are accepted (as was Andrew Martin). These both suggest that the story takes place near the beginning of The Positronic Man.
"Too Bad!": Sometime in the 22nd century, before the creation of the simplified robots in "That Thou Art Mindful of Him".
"That Thou Art Mindful of Him": Takes place about two hundred years after US Robots is founded (i.e. circa 2182). Mentions that the Machines phased themselves out of existence a hundred years earlier (circa 2082).
"Carhunters of the Concrete Prairie": The combination of interstellar exploration and robots places this story at the time of the settlement of the Spacer worlds.
The Positronic Man: Andrew Martin is about a hundred years old "nearly two centuries" after Susan Calvin’s death in 2064.
"Mother Earth": Takes place at the end of "the first few centuries of interstellar travel" when the Outer Worlds "were controlled, politically and economically, by Earth." (quotations from The Caves of Steel, Chapter 5).
The Caves of Steel: Takes place a thousand years after emigration to the Outer Worlds ends. It also takes place 19 years after Elijah Baley first meets Jessie Navodny "back in ’02."
The Naked Sun: Takes place one year after The Caves of Steel.
"Mirror Image": Takes place one year before The Robots of Dawn i.e. one year after The Naked Sun.
"Strip-Runner": Takes place after Elijah Baley starts his Outside group, but shortly before The Robots of Dawn (since the events in that novel aren’t mentioned in the story).
The Robots of Dawn: Takes place two years after The Naked Sun.
Robot City: Takes place about twenty years before Robots and Empire (Han Fastolfe is still alive, and the number of Settler worlds is smaller than in RaE).
Robots and Aliens: Takes place one year after Robot City.
Mirage: Features older versions of characters from the Robot City series, but takes place before the disappearance of the Solarians in Robots and Empire.
Chimera: Takes place one year after Mirage.
Aurora: Takes place one year after Chimera.
Have Robot, Will Travel: Takes place five years after Aurora.
Robots and Empire: Takes place two hundred years after The Robots of Dawn.
Caliban: Takes place a century after the Solarians vanish.
Inferno: Takes place one year after Caliban.
Utopia: Takes place five years after Inferno.
The Stars, Like Dust: Takes place a thousand years after Earth suffers nuclear bombardment (perhaps in an attack by the more conservative Spacer worlds).
The Currents of Space: Takes place five centuries before the founding of the Galactic Empire. (Note: Foundation’s Edge takes place about 22,000 years after interstellar travel begins, i.e. 24,000 CE. This is 12,566 years after the founding of the Galactic Empire, which sets the Empire’s foundation around the year 11,500. For no good reason, I’ve chosen 11,585 CE for the year 1 GE.)
Pebble in the Sky: Stated in the novel.
"Blind Alley": Stated in the story.
Prelude to Foundation: Stated in the novel.
"Eto Demerzel": Eight years after Prelude to Foundaton.
Foundation’s Fear: Shortly after "Eto Demerzel".
"Cleon I": Ten years after "Eto Demerzel".
"Dors Venabili": Ten years after "Cleon I".
"Wanda Seldon": Ten years after "Dors Venabili".
Foundation and Chaos: Same time as "The Psychohistorians".
"The Psychohistorians": Takes place two years before Hari Seldon’s death in 12,069 GE.
Foundation’s Triumph: Takes place after "The Psychohistorians".
"Epilogue": Hari Seldon’s death.
"The Originist": Begins shortly after Hari Seldon’s trial, ends several months after Seldon’s death.
"The Encyclopedists":Takes place 50 years after the Foundation is established in 12,068 GE.
"The Mayors": Takes place 30 years after "The Encyclopedists".
"The Traders": Takes place "two decades" before "The Merchant Princes".
"The Merchant Princes": Begins "almost 75 years" after "The Mayors", ends six years later.
"The General": Begins "over 40 years" after Hober Mallow’s meeting with Onum Barr in "The Merchant Princes".
"Trantor Falls":Takes place forty years before "The Mule".
"The Mule": Takes place 17 years after Han Pritcher joins the Army in 293 FE.
"Search by the Mule": Takes place five years after the end of "The Mule".
"Search by the Foundation": Stated in the story.
Foundation’s Edge: Stated in the novel.
Foundation and Earth: Takes place immediately following Foundation’s Edge.
"Foundation’s Conscience": Stated in the story.
"No Connections": Takes place sometime after the establishment of the Second Empire.