This is an idea that came to me recently, I thought I'd toss the scenario at you lot to see how you think it'd progress.
It's the modern day. A group of aliens are quietly performing mining operations on some large resource-rich asteroids at the edge of our system. Unfortunately, something goes horribly wrong and the entire crew abandon the ship, fleeing the system in escape shuttles. Before leaving the captain set the ship to fly straight into the sun, to ensure it doesn't fall into the hands of the primitive locals. However, after they've left an errant asteroid hits the ship and alters its course - punting it into a rough orbit around Earth.
This ship itself is massive, a full five kilometres from bow to stern. Most of this bulk consists of huge bays to transport the mined resources, with another large chunk taken up by the engines, FTL drive and the reactor. A relatively small part of the ship is given over to crew quarters, medical facilities, a few research labs, etc. Being a mining vessel it's mostly unarmed, with just a few small turrets designed more for shooting down incoming chunks of rock than taking on other ships. However, it does have a "weapon" in the form of a huge mining laser on its belly used to carve up asteroids. The ship also has two hangers, one on each side. They're virtualy empty due to the shuttles being used for the evacuation, but there are still a half dozen or so unarmed shuttles lying around.
The ship is completely depopulated but still has power - enough to last for about a century. The emergency that scared the crew off has been contained by the ship's AI. The AI itself has stabilised the ship's orbit around Earth, and has gone into a dormant standby mode. For the most part if doesn't care about what happens aboard the ship and won't try to stop anyone who happens to wander aboard (that's a job for the now missing security detatchment). Its only concern is to see that no harm comes to the ship, so it will only activate and step in if there's a problem. The AI itself isn't sapient - it's just a program designed to ensure the safety of the ship.
So with the ship itself established, here's a couple of scenarios. What I'm interested in is how you think people and society itself would react.
1) An amateur astronomer is gazing through his telescope one night when he spots the above mining ship just as it enters orbit. After picking himself up off the ground he rushes out to inform every news station and astronomy group he can find the phone number of. By the next day word has spread across the world of the mysterious ship, and observatories in all countries are providing pictures and videos of the vessel. How does society as a whole react to a massive ship turning up in orbit? How do the world's governments react?
2) The space shuttle Discovery is on a routine mission to drop supplies up to the ISS. Just after undocking from the ISS, its pilot looks out the window and promptly discovers the problems of wetting oneself in zero-G when he sees the ship looming above them. The ship has only been seen by the personel on board the Discovery, and NASA is the only agency that knows if its existence. It'll be two months before anyone else discovers the ship's existence. How would the US's government react to the find? Would it attempt to board the ship? Would it try to suppress the ship's existence becoming public knowledge?
Finaly, in both scenarios, just what would be the result of trying to investigate the ship? Assuming people manage to get aboard it, how long would it take humans to figure out how to understand the completely alien language that everything is written in?
"You've all been selected for this mission because you each have a special skill. Professor Hawking, John Leslie, Phil Neville, the Wu-Tang Clan, Usher, the Sugar Puffs Monster and Daniel Day-Lewis! Welcome to Operation MindFuck!"