A normal cargo with a strange buyer/receiver
A strange cargo
Animal cargo
An annoying passenger
A passenger you're sorry to see go on their way
A hijacker
A VIP passenger with bodyguard and retinue (everybody look sharp)
Transfer of a data satchel with important info (i.e. a single isolinear chip, data solid, or such)
Engine problems
An offer to upgrade the engines (or other systems) but for a steep price/trade/favor
Orion Pirates want your cargo (and your Captain!)
A smuggling offer
A chance to bid on a Federation gravy run taking supplies to a Starbase
Cryogenically frozen Orion females (unknown to the crew, of course)
Attacked by a Klingon raider
Sold into slavery
That should give you some ideas. Half a million ton cargo freighters can cross an ocean with false info on their manifest, in real life. And that can range from illegal drugs to firearms to human trafficking. Then there's the legal stuff in a container, which could be anything and everything.
You're always going to get similarities (and comparisons) to Firefly just because the ship has the same purpose. The trick is making it look/sound 'Star Trek-y'.
Your ship will be going from point A to point B on a Starmap. You have to tell the story about how it did or didn't get to Point B.
