There are six people in each town that you can help and/or impress in order to gain their trust. For each eye, you need the approval of four of them. Alternatively, you can challenge the keeper of each eye, and if you win their challenge, they will give the Eye to you.
Here are the people whose approval you can get:
Eye of Fornsigtuna
Volund:
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×he already trusts you because you have saved his son.
Lithrasir:
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×You must reunite her with Liff
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×Deliver her letter to Liff - find her flowers on Liff's behalf - get rid of the troll on the bridge to Munarvagir - agree to escort Lithrasir to the holt at nightfall
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×Eitri the Svartalf has the flowers you need, he will ask something in exchange for them.
Snorri:
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×Win at an archery contest against him (there's a dedicated thread with tips on how to do it)
Arngrim:
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×the old guildmaster lives in the watchtower because he is exiled. Find a way to cure his sickness so he can return home.
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×You will need to reunite the lovers first
Aurvandel:
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×He has posted a task on the guild board, he'll trust you if you accomplish it
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×Read the Adventurer's Book about how to defeat a haugbui
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×You will need to go to the graveyard at night to discover which is the haugbui's grave, then dig out its bones, burn them to ash, and scatter them to the winds
The Jarl:
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×He has gone missing. Find him, and bring him back safely.
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×He is enthralled by the huldra. find a way to deflect her hypnotic gaze.
Eye of Munarvadir:
Sigurd:
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×He has forgotten someone important to him. Find a way to restore his memory.
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×You will need to challenge the Svartalf Skrymir, and win.
Liff:
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×You need to reunite him with Lithrasir. See above.
Kraka:
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×She's in jail in Fornsigtuna. Help her out.
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×Either save the Jarl and ask for her release as your reward, or bribe Snorri, or bust her out yourself.
Sigrun:
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×She really misses her dead husband. She will give you the key to his crypt if you click on it.
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×Eitri the Svartalf has a magical artefact that lets you speak to the dead. Trade something for it, like a cat's footfall
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×You will need a sheet of paper, some ink, and a cat.
The librarian:
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×His cat is missing, find it for him.
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×The cat is in the frozen garden. You'll need to lure him down from the tree, or to catch him before he climbs.
Regin:
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×He has unfinished business with his brother, Fafnir. Bring him his father's chest as proof that you've dealt with his brother.
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×The chest is hidden in the crystal cave in Svartalfheim. Very well hidden.
That's a broad overview of how you gain people's trust. I've tried to give as many hints as possible without giving too much away. If you're really stuck on a particular task, I'll be happy to provide a more detailed explanation, but you should try to figure out things for yourself first.