The $64,000 question is:

Where is Equestria, in relation to the Federation/Klingon empire/Gamma or Delta quadrants?

Well, the official MLP:FiM show has given us a few clues over the years:

1. From the song "The Art of the Dress", we know Equestria has a constellation in its sky called "Orion", and that this constellation has 3 stars in its belt.

2. From the same song, we also know that Equestria has constellations named Canis Major and Canis Minor in its sky.

3. From the first Great-and-powerful-Trixie episode, there are strong hints that constellations called Ursa Major and Ursa Minor also exist in Equestria's sky.

4. In "Hearth's Warming Eve", ancient Pinkie Pie tells ancient Applejack that "the Earth is round", referring to the planet they were standing on (said planet being the eventual place of Equestria later in that episode).

The conclusion is obvious.
Equestria is in Sector 0,0,1

2101273
I've got a problem with #4 here. The reason that Earth is called earth is because That is the English word for the ground we stand on. That naming convention could have been adopted by any number of cultures quite easily, because it's an easy link to make. Many species would refer to their home world in a similar fashion.

As for 1-3, Those are again based more on names than actual facts. Unless we can verify that all of these constellations are the exact same shape as the ones one earth (we can only do that for the Ursas since we did see those in the animation), we can't determine that these are the same constellations as the ones in Equestria, despite the high likelihood that they are. There's also the fact that constellations are completely subjective and don't have anything to do with actual locations.

The only things that are absolutely certain about Equestria are that it's a class-M planet, has strikingly similar vegetation and animal life to earth, and that the planet it is located on has a higher gravity than it's sun since it's shown from the first episode that the sun is in an orbit around the planet, whether or not Celestia is moving it.

It seems plausible, but Starfleet has already explored all of Sector 001 by the beginning of the original series. Unless Equestria's home system was shielded by being inside a nebula, they would have found them.

But if we ignore all the television episodes and look behind the scenes, it is possible that Starfleet could have discovered Equestria, probably during Archer's era or by Archer himself. If so, that would lead to a pretty interesting encounter, and one hell of a backstory.

2101273 In my story Voyager comes to Equestria I plan on having it that -SPOILER ALERT!- the Caretakers, Q and the Greek Gods that Kirt encountered in an old episode of Star Trek are all from Equestria and the greek gods made the night sky on Earth look like the night sky in Equestria.

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WHAT DO YOU MEAN WE CAN'T GO TO EQUESTRIA FOR SHORE LEAVE

In one brainstorming/RP session for a possible ST/MLP tale, we had the Equis system near the edge of Federation space, in a shell formed by a dark matter nebula created by discord.
No light goes in, none comes out, with the stars inside the shell simply projected against the backdrop as if seen from the Earth. No one had bothered to explore it because the technology star ships rely on for crew survival dont work right around dark energy/matter... until a certain Draconequis decides to have some fun.

Its obiously in the horsehead nebula.

2102300 i like your theory. on the other side of a black hole and the weird physics bending gravity allows the sun and moon to be controlled

What if Equestria is in a wandering star system? If the sun moves around the planet (or Celestia and Luna move the planet and moon around to better get light from the sun), the star wouldn't necessarily have to orbit the center of the Milky Way Galaxy like other star systems.

In fact, if Celestia can move the sun (however accurately or for how long) the Equestrian star system could function more like a giant spacecraft. If the Equestrians want to visit the Alpha Quadrant, they just nudge their sun in that direction, move their planet and moon accordingly (dragging along any other planets they feel necessary) and sit tight while their whole star system heads for their destination.

Granted, they might not be traveling at warp speed (or even relativistic speed), but Celestia and Luna are thousands of years old and could have been 'piloting' this thing all that time. It would actually make sense to a degree given all the work the ponies have to put in to keep their ecosystem going... they are all crew on a giant planet-sized spacecraft which looks pretty much exactly like an actual planet.

Luna might have an important job with navigating or handling incoming meteors or otherwise maintaining all the celestial bodies that aren't the Equestrian planet. If their star system has an asteroid belt, then Luna would have to make sure all the asteroids are kept in position if they ever need to do 'course corrections'. Or she has to keep an eye out for incoming meteors, space ships, or keep a gas giant or two around to act as a buffer in case they need to intercept something.

In short, when Celestia and Luna can move the sun and moon around, all bets are off in regards to where their star system is because they could potentially move it wherever they want to. The only question is how quickly their star system can safely accelerate, slow down, or provide course corrections.

It would actually be interesting if Celestia and Luna kept an eye out for stars in danger of going nova and checked them out to see if they can help. Either Celestia checks the star to see if she can 'repair' it, or if there is an inhabited planet, they discretely maneuver the Equestrian star nearby, move the new planet into orbit around their sun, and then fly off to see if they can drop their new friends off in a stable orbit around a healthy young star somewhere.

2105264 The only problem I have with a "wandering equestrian star system" is the mentions in the show of the constellations. If Equestria's sun really were moving through interstellar space at the kinds of speeds you're talking about, the constellations would visibly shift in the sky from one year to the next. Surely, Twilight Sparkle wouldn't insist on the number of stars in Orion's belt if the "belt" kept on moving!

2102300 Your "Equestria is hidden in a nebula with projected stars" scenario would explain an awful lot. Like how Celestia "raises the sun" in the morning. I've always thought Celestia was merely altering the rotation rate of Equestria, and not actually moving the sun itself -- but if they're in a Discord-created shell, who the heck knows HOW big the sun is relative to Equestria! Discord might've made the sun a few miles across and whiz by a couple hundred miles above Equestria's surface, like the Flat Earthers believe.