2D real-time spaceship adventure games
As much as I like this particular niche, I've only played a few of the games listed. If something is missing, or one I've added doesn't belong - which is likely - please let me know!
What is here? Like the title says, games where the player controls a spaceship in real time on a 2D plane (and hopefully has an adventure!). No 3D movement, no turn-based space movement, and at least some attempt at plot or role-playing-ness - i.e. no shmups, and probably few twin-stick shooters and multiplayer games.
Almost-suitable games: Star Nomad Elite (not on PC), Starport: Galactic Empires (MMO), Star Sonata (MMO), Scum of the Universe (arcadey), Fusion: Genesis (not on PC, mostly multiplayer twin-stick shooter)
I had a hope that making this list would help me find a game I barely remember from perhaps a decade ago. While it hasn't turned up yet, I've found maybe half a dozen more that do belong, so time spent looking hasn't been wasted.
Update, many years after I originally made this (probably with a bunch of new games now deserving to be on here): the game I wanted to find was Starfight VI: Gatekeepers.