In Star Trek: First Contact, Riker comments how strange the moon looks and that in his time you can see the city of New Berlin. In TNG 'The Naked Now', Data begins reciting a limerick about a girl from Venus. I acknowledge that a limerick is no proof of colonisation on Venus, but it got me to thinking about what celestial bodies in the Sol system have been colonised. For the purposes of this question, I define a celestial body as a planet, moon, comet or asteroid.
I did a google search restricted to "site:chakoteya.net" (a site with transcripts of every Trek episode) along with searching for each planet, I found the following:
Mercury: No scripts suggesting outposts on Mercury.
Venus: In the Voyager episode "Future's End", Chakotay says that after he trained as a pilot in North America, "I went to Venus for a couple of months to learn how to handle atmospheric storms, and then I dodged asteroids for a semester in the belt." It's possible this just involved living on a vessel temporarily parked in the vicinity of Venus, or a space station in orbit around it, though. And in the DS9 episode "Past Tense Part 1", O'Brien reports losing contact with all parts of our solar system including "the terraforming stations on Venus", though again we can't be sure if these are orbiting stations or stations on the surface.
The Moon: In the movie First Contact, Riker tells Zephram Cochrane that "There are fifty million people living on the moon in my time. You can see Tycho City, New Berlin, even Lake Armstrong on a day like this." A bunch of other references to life in lunar colonies can be found in the Luna article on memory alpha.
Mars: Definitely colonized, in the TNG episode "The Drumhead" there is a trial in which the prosecutor, Satie, says to a witness "Mister Tarses, your records state that you were born on Mars Colony." There have also been numerous references to starships being built at Utopia Planitia on Mars (first mentioned in the TNG episode "Booby Trap"), apparently they maintained a large shipyard there, and you can see from this shot (from the TNG episode "Parallels") that it was located on the surface rather than in orbit. In the Voyager episode "The 37's", Kim says "Mars was colonised by people from Earth in 2103."
Asteroids: In the Enterprise episode "Terra Nova", Archer discusses the first attempt to create an interstellar colony, saying "Could humans colonise deep space? They'd already build New Berlin on the Moon, Utopia Planitia on Mars, even a few asteroid colonies, but all within our solar system. When they found an Earth-like planet less than twenty light-years away, it was hard to resist." He says that they lost contact with the interstellar colony and hadn't heard from them in "over seventy years", so there must have been asteroid colonies more than 70 years before the date the episode was set, which was in 2151 according to this page on memory alpha, so the colonies would have been established before 2081 (which itself was less than 20 years after the invention of warp drive in 2063, so perhaps they were initially established during the pre-warp years). Of course this doesn't tell us whether they still maintained asteroid colonies in the TNG era but it seems likely. Note that the reference to Utopia Planitia seems to contradict Kim's statement that Mars was colonized in 2103, but the Mars article on memory alpha mentions that in the book Star Trek: Star Charts, 2103 was retconned into the date that "the United Martian Colonies, the uniglobal government of Mars, was founded."
Jupiter: A gas giant so presumably no surface colonies, but the TNG episode "The Best of Both Worlds, Part 2" has the line "The Borg have dropped out of warp, sir. Jupiter outpost nine two reported visual contact", and in the DS9 episode "Doctor Bashir, I Presume", Lewis Zimmerman (the basis for the Emergency Medical Hologram on *Voyager) says "I'm Lewis Zimmerman, Director of Holographic Imaging and Programming at the Jupiter Research Station." Jupiter Station was also shown in the Voyager episode "Life Line", in which Reginald Barclay met with Zimmerman there, and it apparently already existed by the Enterprise era since Archer referred to "The armory team at Jupiter Station" in "Silent Enemy". The TOS episode "By Any Other Name" also featured Scotty bringing out some green liquor and saying "I found this on Ganymood, er, Ganymede"--Ganymede is a moon of Jupiter, so unless there was another colony or planet by the same name, this probably indicates it was colonized.
Saturn: Another gas giant. No clear mentions of colonies, but in the TNG episode "Chain of Command", Captain Jellico says "I began my career as a shuttle pilot, on the Jovian run. Jupiter to Saturn and back once a day, every day." Probably there would be no use for regular trips between Jupiter and Saturn unless there was some kind of outpost in the vicinity of Saturn. The TNG episode "The First Duty" also discusses in accident flying in the vicinity of Saturn and features the line "We had less than two seconds to activate our emergency transporters and beam to the evac stations at Mimas"--Mimas is a moon of Saturn, so apparently there was some kind of outpost there.
Uranus: Didn't find lines indicating colonies or outposts there.
That's it for stuff that's definitely canon. Beyond that, the memory alpha articles mention the following info from books:
According to Star Trek: Star Charts (p. 19; "United Federation of Planets I"), Mercury was classified as a B-class planet. This planet was a charter member of the United Federation of Planets in 2161. ... In William Shatner's novel Captain's Glory, Mercury was home to a research facility, stated to be one of the oldest outposts in the Sol system.
According to Star Trek: Star Charts (Pgs. 38, "United Federation of Planets I"), Venus was classified as a N-class planet. This planet was a charter member of the United Federation of Planets in 2161. In the late 24th century, a major orbital facility was the Aphrodite Terraforming Station.
In the Fifteenth UK Story Arc, set in the late 2260s, a research depot on Venus lent the USS Enterprise experimental repair robots.
According to Star Trek: Star Charts (pp. 32, "United Federation of Planets I"), The Moon (or Sol IIIa) was classified as a D-class moon. In 2039, the Lunar Colonies, the uniglobal government of the Moon, was founded. Tycho City was the seat of government. In 2161, the Moon was a charter member of the United Federation of Planets. In 2378, Humans were the dominant species, and there were counted 50.2 million living on the Moon. Points of interest included Tranquility Base, Lake Armstrong, New Berlin, and Lunaport.
According to Star Trek: Star Charts, (p. 32 & "United Federation of Planets I"), Mars (or Sol IV) was classified as a K-class planet. In 2030, Aries I, the first manned Earth ship landed on the planet. In 2103, the United Martian Colonies, the uniglobal government of Mars, was founded. Utopia Planitia was the seat of government. The first flag of the UMC was based on a painting of a bullfighter on velvet. In 2161, Mars was a charter member of the United Federation of Planets. In 2378, Humans were the dominant species on the planet and there were 133.8 million inhabitants living on Mars. Points of interest were Olympus Mons, Valles Marineris, and the Utopia Planitia fleet yards.
According to Star Trek: Star Charts (Pgs. 22, "United Federation of Planets I"), Jupiter was classified as a J-class planet. This planet was a charter member of the United Federation of Planets in 2161.
According to Spaceflight Chronology, the Earth-Saturn probe mission was conducted on the UNSS Lewis and Clark. Signs of past mining activity was discovered from the second and fourth moons of Saturn, first proof of the existence of extraterrestrial intelligent life. This last detail and "One Small Step" would put the mission after 2032. ... According to Star Trek: Star Charts (p. 39, "United Federation of Planets I") and StarTrek.com , Saturn was classified as a J-class planet. This planet was a charter member of the United Federation of Planets in 2161.
According to Star Trek: Star Charts ("United Federation of Planets I"), this planet was a charter member of the United Federation of Planets in 2161.
According to Star Trek: Star Charts ("United Federation of Planets I"), Neptune was a charter member of the United Federation of Planets in 2161.
According to the Spaceflight Chronology the first Pluto Base began operation in the year 2039 as a navigational check point for extra-solar missions. According to The Worlds of the Federation (p. 12), Pluto was the site of the Pluto Research Base, where all life was decimated by a strain of the omega virus, some time after the founding of the Federation. According to Star Trek: Star Charts (Pgs.39, 62, "United Federation of Planets I"), Pluto was classified as a C-class planet. The planet was a charter member of the United Federation of Planets in 2161.
Additional info from novels, role-playing game modules and such (stuff not covered by the memory alpha site) can be found in the links in the Sol article on memory beta.
On the canon scale, this answer is pretty low. But it exists, so here you go.
In the video game Star Trek: The Next Generation: Birth of the Federation, which was hurriedly released in 1999 to try to stabilize MicroProse's failing business, the Sol system is the starting system for the Federation player. In the game, all rocky planets are able to be eventually colonized. However, the concept of moons, asteroids, or planetary stations does not exist in game. While you can build an outpost or star base (simply an upgraded outpost), you can only build one for the entire system. Given that information, then Mercury, Venus, Earth, Mars, and Pluto are inhabitable or can be terraformed to be inhabited.
Here's a screenshot of the Sol system in-game. Blue lettering indicate gas giants (cannot be terraformed or colonized), red lettering indicates planets that can be terraformed but have not, yellow lettering indicates terraforming is in progress, and green lettering indicate that the planets are habitable. M-class planets are inhabitable without terraforming. Starting the game with advanced technologies (Sovereign and Defiant ships are available at the start), all 5 rocky planets in the Sol system, including Pluto, are inhabited.