These are the miscellaneous quests that can be done any time, regardless of your standing with the guilds.
Separated At Birth
Brush With Death
Seeking Your Roots
This quest is started when you pick up any Nirnroot. While the quest gives you many markers to follow (each leading to the Mages Guild of a different city), the one you want to go to is Skingrad. There, you will learn of an alchemist that works in the basement of a tavern (also in Skingrad). Show him the Nirnroot, and he'll tell you about it as well as inform you that it's the essential ingredient in a very useful potion.
The remainder of this quest consists of finding Nirnroots and giving them to the alchemist, to receive increasingly powerful variants of this potion.
Unfriendly Competition
Talk to Jensime in the Imperial City (she runs Jensime's "Good As New" Merchandise in the Market District). She will make a passing comment about a guy named Thoronir. Ask her about him, and she will explain that Thoronir's prices are absolutely insane, and that she's certain that his merchandise is stolen. Agree to help her, and wait outside The Copious Coinpurse (Thoronir's shop) until he leaves. Follow him around for a while and you'll witness him meeting with a suspicious Nord named Agarnir.
After the meeting, start following Agarnir, and you'll eventually end up at his house. Start waiting until Agarnir leaves, then sneak into his house (making sure the guards don't see you). Go down and search the basement until you find what you're looking for.
I can't remember the rest of this quest.
Whom Gods Annoy
Go to Leyawiin and listen to rumors until you hear about a woman named Rosentia Gallenus who has been holed up in her house for weeks with an awful stench and odd animal noises coming from it. This will give you the quest marker that leads to her house. Talk to her, and she'll tell you about a curious Daedric staff that she purchased some time ago. As it turns out, it's a fake, and it has the humiliating side effect of summoning four scamps that follow the carrier wherever they go. Rosentia will point you to the alchemist at the Leyawiin Mages Guild. Talk to her, and you'll learn that the Staff of the Everscamp can only be rid of if somebody willingly accepts it from the current bearer, and that it can be disposed of at last at Darkfathom Cave (north of Leyawiin). Talk to Rosentia once more and ask her to give you the staff. Then make your exit and head for Darkfathom Cave.
Darkfathom Cave is a fairly straightforward dungeon. You'll probably end up fighting Scamps and other Daedra (higher level characters may encounter Fire Atronarchs). Once you make it to the altar, drop the staff (it doesn't matter where as long as the quest log updates with the message "I no longer have the urge to possess the staff") and book for the exit. Talk to Rosentia once again, and she will give you the Ring of Eidolon's Edge, which fortifies your Blade skill.