26th century

Multiple realities

(covers information from several alternate timelines)

The 26th century was defined in the calendar of Earth as being from 2501 until 2600.

Decades[]

The Federation has become an even more powerful entity, with species such as Klingons, Ithenites, and Xindi among its citizenry. (ENT: "Azati Prime")
A time-travel pod travels to New Jersey in the 22nd century and then to the USS Enterprise-D in 2368. (TNG: "A Matter Of Time")
Had the Kelvan invasion force in the Milky Way Galaxy remained in control of the USS Enterprise, it was theorized that the starship would arrive in the Andromeda Galaxy sometime during this century. However, the Starfleet crew regained control of the vessel, forged a cooperation with the Kelvans, and launched a robot ship towards Andromeda proposing an alliance between the Federation and the Kelvan Empire. (TOS: "By Any Other Name")
Had the charge against Mogh, for aiding and abetting the Romulan attack on the Khitomer Outpost, been upheld by the Klingon High Command in 2366, the family name of the House of Mogh would have been disgraced for seven generations, with the family getting its first chance of returning to grace in this century. (TNG: "Sins of The Father")
If the statistical calculations of Julian Bashir's colleagues from The Institute concerning the unstoppable Dominion conquest of the Alpha Quadrant had been accurate, then the hypothetical rebellion against Dominion rule would have occurred during this century, assuming an average length of about 30 years for a given generation, plus or minus five years. (DS9: "Statistical Probabilities")

2507: The Terran Empire would have succumbed to the galactic revolt predicted by the Halkans, in the approximate prognosis of the mirror universe Spock, had it not been overthrown prematurely by the Klingon-Cardassian Alliance. (TOS: "Mirror, Mirror")

2553: The Federation is claimed to have destroyed the Xindi homeworld in this time. (ENT: "The Expanse")

2554 (approximate): In the future shown to Jonathan Archer by Daniels, the forces of the Federation defeat the invasion of the Sphere-Builders in the Battle of Procyon V and force them back into their own trans-dimensional realm. The starship USS Enterprise-J participated in this battle. (ENT: "Azati Prime")

Daniels said the battle took place "four hundred years in the future" from 2154, but did not give an exact year.

2562: A neutron star in a binary star system in the Kavis Alpha sector explodes. (TNG: "Evolution")

2568: A robot ship launched to the Andromeda galaxy from the Milky Way galaxy directed to Kelva is expected to arrive in this year. (TOS: "By Any Other Name")

2569: Fred sees a tan zhekran for the last time until shortly before his death in 3191. (DIS: "Red Directive")

2573: A component that will be used in the Xindi destroyer/probe is manufactured. It is sent to the Xindi in the 22nd century by the Sphere-Builders. (ENT: "The Expanse")

2590 (approximate): The last known contact between the United Federation of Planets and the Q Continuum occurs. (DIS: "The Examples")

Star Trek: Final Frontier was to be set in this century, specifically the year 2528.
Going by Data's statement that the energy ribbon from Star Trek Generations passes through the Milky Way Galaxy every 39.1 years, it can be expected to return in 2527 and 2566.