The game genre you are describing is often referred to as "rail shooters" since whether you're a person, spaceship or other vehicle, the movement is usually completely controlled by the game and you therefore move along a pre=scripted path as if on a rail while the player merely aims the cursor to shoot targets. There are a bunch of games in this genre on the dreamcast, stuff like Virtua Cop, Confidential Mission, House of the Dead, Rez and others, but I can't think of any spaceship/space themed ones.
There are tons of spaceship related rail shooters and similar games that don't give you much control over the spaceship on other 90s consoles including the 3DO, Sega CD and PS1 though, there might be some on Saturn as well. I could help throw some names out there if you can provide some additional details such as whether the game had polygon graphics (textured or untextured) or sprites, whether you were in the cockpit of the ship or viewed the ship from a chase view, whether every level you remember took place in space or whether there were levels that took place on planet surfaces.