How about a bastard child wanting to track down a noble parent, ostensibly for that "where do I come from", but actually the "child" is an assassin using the PCs as a way to get close to his/her target. Or maybe trying to establish claim to the noble's title and planning to bump off the "legitimate" heirs.
Guild war: two petty guilds are trying to gain control of the street from the other. "Innocents" are getting caught between them. The PCs can decide which will win, or wipe them both out.
An actual honest and true paladin is looking for her brother, a beneficent hospitaler who went missing in the dregs. The brother has been kidnapped by a guild, and manipulated into helping heal their enforcers, even during the middle of bloody skirmishes. The brother's life is in danger, and his faith and power are being abused and corrupted.
A group of youngsters have just sold their souls for power, have just come of age and are lashing out at everyone that ever hurt any of them. Build a party of different warlocks with maybe a Death Priest for support & healing. The party has to calm them down or kill them.
Likewise, a youngster has made a deal with an extraplanar being, but changed his/her mind. The Pact is due to become binding in 24 hours. Can the party find the being with the contract and get her released (by hook or by crook). Maybe this youngster just helped the party in some way (stopping someone from escaping, warning of an ambush, what have you), and asks the party to help break the contract as a reward.
Street urchin asks the party if they need a guide for 3 pieces of gold. If they turn the urchin down, have a couple short fights hit them (lvl-2 easy stuff, but annoying and draining resources) and a pick pocket go after them (don't take anything vital--let the party chase the cutpurse down and recover the property). After the second one, a different urchin asks if they need a guide. If they ever accept a guide, all such activity stops, as the guide gives subtle signals to the riff-raff to back off.
Find a scene of carnage, where the street fight just ended, and the beggars are scrambling to loot the bodies. Maybe even fighting each other.
Slave auction. "How much am I bid for this fine goblin? See her teeth? This one is a prime breeder, guaranteed to improve your herd. Look at these muscles. She can do plenty of work too. And these claws are good if you just want something to throw into an arena. Now, this young kobold has been gentled to act as a house servant. You'll get no trouble with this one, so long as you keep a good lash. Just in: this batch of svirfneblin. They may smell bad, but they are good brute labor, so long as you have a sharp whip and sharper eyes. Don't let those pleading eyes fool you--these are no surface gnomes..." (but actually, they are, captured in a surface raid 3 days ago, starved, and forced to wear actual svirfneblin rags. Their tongues have been branded, which hampers their speech until they can get proper care, or at least 3 months have gone by.