Here's the classics:
-Kidnap a corp employee so they can work for another corp.
-Rescue a corp employee who no longer wants to work for a corp or is being held against their will.
-Investigate who killed someone. (Your friend, a relative, a friend of a friend, ect...) (This is like, the quintessential SR adventure, because it usually ties into a larger plot.)
-Infiltrate a corp either physically or digitally (or both) to steal sensitive data or destroy an asset.
-Investigate the appearance of really nasty spirits (usually turns out they were sent by someone for a specific purpose.)
-Provide security to a VIP.
-Stop a rogue AI that's run amok.
-Piece together what happened before you got amnesia. (The basis for the SR SNES game, but that was a single player game.)
-Berzerk cyborg killbot is on a rampage and must be stopped.
-Betrayed! (Whatever someone was having you do, the real reason is they need a patsy and that's the party.)
Those basically encompass all the SR games I've played in. Of course, those are the broad strokes. There's usually at least 1 to 3 other plot threads woven into the overarching plot. Bait and switches are often a very popular way of running SR games, since no one is ever completely upfront about their motivations for needing Shadowrunners. (Plausible Deniability, Need To Know, Trust No One, ect...)
Last game our team was hired by a Lone Star detective who wanted us to investigate a case "off the books" because his superiors had told him to drop the investigation, forcibly. The case revolved around another Lone Star detective who was killed while investigating the disappearance of known BTL addicts (Better-Than-Life chippers.) Turns out the BTL addicts were part of some experiment being run by Ares or another corp (can't remember atm.)
We had highly-trained kill teams trying to take us down as the powers that be discovered we were looking into the matter. Investigation led us to an advanced lab out in the Seattle wastes, innocuously disguised as a ruined building. There was a highly trained medical staff and personal security and some spirits involved. But we never really got down to the specifics of what was going on, because my team went in guns blazing and made a mess of things. We definitely found out who killed the detective (found his body at the bottom of a waste shaft in the facility, being snacked on by ghouls), so we technically completed the job, but we never found out the why of it all.
SR can be a lot of things but it's usually a combination of gum shoe detective work and gun battles, with hacking and magic added to the mix. Otherwise, it tends to be about high-end corporate espionage and assassination. As always, these are broad gameplay themes, there's no reason you can't run a SR campaign where the party is a biker gang looking to claim turf, or a group of Shamans and Mages trying to right a great magical wrong or something even more fantastical. A party built as an Urban Brawl team is also a fine way to start things off. Consider running an adventure outside of cities and urban zones to achieve a different narrative effect or to get away from standard SR tropes. (Note, life outside urban zones is very dangerous.)
If you're struggling to come up with an original idea, just bear this in mind: People hire Shadowrunners because they're disposable and they don't mind getting their hands dirty. When "law enforcement" won't help you or your enemies are too powerful to fight in the open (i.e. the corps), you turn to Shadowrunners. Because their ability to live off the grid and SINless means they can take risks and do things other citizens can't. Their cavalier disregard for property rights and corporate extraterritoriality laws means they make good couriers, hitmen, spies and of course....fall guys!