The Museum of Oddities
Where: Crucible, Museum of Oddities (C5);
The museum needs new exhibits. Every time you find something odd enough to be on display here, you'll get an appropriate journal entry. The proprietor pays you 200-300 Gold for each item. Most of the oddities are unfortunately found in completely random places: as loot, contents of chests, or when collecting alchemical ingredients. There are 12 items to be found in this quest, but only locations of 2 of them can be described.
Pelvis of Pelagius can be found when doing the Symbols of Office quest. Check out the main quest guide for details.
Din's Ashes are on a hidden level of Ebrocca (SI29). To get there, you have to find a hidden button under one of the leather shields not very far from the main entrance (screenshot below). Ashes are in one of the urns there.
Finding any other, random oddities will require some luck. Here's the list:
Ring of Disrobing
Dagger of Friendship
Blind Watcher's Eye
Deformed Swamp Tentacle
Mute Screaming Maw
The Antipodean Hammer
Where: Bliss/Crucible, Cutter/Dumag gro-Bong;
Now, this is listed as a quest in your journal, but there's really nothing to be completed here. You just find out that blacksmiths' in Bliss (Dumag) and Crucible (Cutter) can forge Amber (found in Mania) and Madness Ore (found in Dementia) into new types o weapons and armor. Additionally, Amber is a random loot left by Gnarls, and Madness Ore is often found by dead Grummites.
Besides, from time to time you'll come by an item called the Matrix - these are elements that can add magical abilities to forged weapons.
Amber is forged into Light Armor, Madness Ore into Heavy Armor.
A Liquid Solution
Where: Sickly Bernice's Taphouse (C16);
The proprietor of Crucible's inn is very sick and needs your help. You have to go to Knotty Bramble and collect a cure for her - it's called Aquanostrum. Head to the cave that's been marked on your map. The medicine is found on the second level. The dungeon is crawling with Grummites - it's a good opportunity to collect some Madness Ore. When you grab the Aquanostrum (you just have to approach the statue in the middle of the pond), go back to Bernice. She'll give you a Circlet of Verdure - it gives you 50% poison and disease resistance, as well as an increase in health and endurance.
Where: Crucible, Ushnar gro-Shadborgob;
During the day, Ushnar can be found in the central part of Crucible. He'll ask you to get rid of a beggar called Bhisha. You can pay the poor fella to leave (100 Gold), kill him, or raise his disposition high enough so that you can convince him to go away. Your reward for this quest will be a disgusting little doggy, who'll aid you in combat.
Where: Crucible, Earil (C7);
Earil wants to get rid of a thief named Brithaur. The little crook doesn't want to leave or change his evil way until you bring him 5 Flawless Pearls. You can try to find them and bring them to him, but it isn't really worth the trouble. Just kill the bosmer and take his heart to show Earil that you've solved the problem.
The Coming Storm
Where: Crucible, Things Found (C13);
Ahjazda, the proprietor of Things Found in Crucible wants you to bring her some items. She needs the Amulet of Disintegration, the Ring of Desiccation, and Calming Pants (must have the calming pants!)
Ring of Desiccation is on display in Museum of Oddities - just steal it then the lady's not looking.
Calming Pants are owned by Fimmion, an elf wandering around Bliss. He'll gladly part with them if you give him a Sweetroll. If you don't have the patience to look for it, just kill him and take the loot.
HINT: You can find a Sweetroll in Pyke's house in Hale (SI16).
Amulet of Disintegration is in the ruins of Milchar. A map marker will show you a shortcut - use it and you won't have to go through the whole complex. You'll find yourself in a shrine with a frozen container in the middle. To get it open, you have to light the three bonfires, using the enchanted torch (you can't equip it, so just hold it by pressing 'Z' on your keyboard). You have to be quick about it - the fires go out faster than you'd expect and they have to be lit simultaneously. Grab the amulet and go back to Ahjazda. She'll teach you a spell called Ahjazda's Paranoia.
The bonfires you have to light.
HINT: After performing the quest, you can kill Ahjazda and give the Ring of Desiccation back to the museum.
Where: Crucible, Hirrus Clutumnus;
When you find Hirrus on the streets of Crucible, he'll ask you to meet him after dark. Wait until nightfall and go to the map marker. He tells you that he wants you to kill him in a sneaky way. Agree and wait until the next day. Find him standing near the edge of high stairs - talk to him and push him down. Take a key from his body and go to his house. You can find a Ring of Happiness in there. How ironic.
Where: Bliss, Amiable Fanriene
Despite being quite wealthy, Amiable lives on the streets of Bliss. That's because he's afraid of walls. You are to find him a safe spot to sleep. Talk to one of the beggars and you'll be told to speak with Uungor. You can find him in the temple during the day. If you raise his disposition high enough, he'll agree to switch beds with Amiable. Your reward for this little quest is a Burst of Power scroll.
The Fork of Horripilation
Where: Bliss, Big Head;
An argonian called Big Head is searching for the Fork of Horripilation. In order to find it, you have to talk to Bolwing. The beggar can be found in the southern part of Crucible. He'll tell you to go to Longtooth Camp (SI9). Defeat some zealots there. The fork is in the jewelry case marked on the map. Take it back to Big Head, and he'll reward you with a skill point (to either Blade, Sneak, or Alchemy).
Work is Never Done
Where: Bliss, Tove the Unresftul (B15);
Tove the Unrestful wants you to help him build his flyboat. He needs calipers and tongs to finish his work. To finish this quest, you have to bring him 100 items total. He'll pay you 5 Gold for each one. If you gather enough of them, take your money from him and the quest will be done.
HINT: If you want to have this done quickly, try battling trolls of Tidewater Cave (7S1, Cyrodiil).