This is a lovely idea for a thread, especially asking for a little bio of the ship history. I think that that sort of world-building makes the creative process much more fun, and I know it has helped hone the look of the ships I have managed to complete.
I've only got two ships in my Star Trek 'fleet', but I hope to change that this year when I get my PC working again. In the meantime, this is the ship with the most backstory gone into it:
This is Reliant, humanity's first (barely) practical interstellar vessel. It was hastily built for a mission to Proxima Centauri, not long after First Contact. Something of a Frankenstein vessel, its sublight engines and tankage were pilfered from a pre-WW3 Lunar Freighter, and its warp drive was powered by a fusion tokamak instead of a matter/antimatter reactor. The landing craft were re-purposed WW3 intercontinental troop shuttles.
En-route, the crew would be in suspended animation for a duration of three years. There were enough consumables for a six month survey of the Proxima Centauri system, before heading home again.
Whilst it was away, spaceflight technology had improved to the point where Reliant became obsolete, and even if that wasn't the case, many of its older components were no longer up to the rigors of interstellar flight. Reliant spent the remainder of its career in the less strenuous role of technology demonstration platform. It mostly flew sub-light in the outer solar system, with brief FTL hops into the Oort Cloud and back.
It presently resides in the fleet museum, as part of the pre-Federation annex.