Starfleet offered William Riker command of the Melbourne in 2366. Even though Jean-Luc Picard described the Melbourne as "a fine ship", Riker declined the promotion to remain aboard the USS Enterprise-D. (TNG: "The Best of Both Worlds")
This ship, along with the Excelsior-class USS Melbourne, joined the Federation task force taking a stand against the Borg cube at the Battle of Wolf 359. Both Melbourne ships, along with 37 other ships, were destroyed. (TNG: "The Best of Both Worlds, Part II"; DS9: "Emissary")
It was never established on-screen why Starfleet sent in both the Excelsior-class and Nebula-class Melbournes to the Battle of Wolf 359 or why both ships seemed to be in service at the same time.
- USS Melbourne (Nebula class) article at Memory Alpha, the canon Star Trek wiki.
- USS Melbourne (Nebula class) article at Memory Beta, the non-canon Star Trek wiki.