Learn Hardcore Icelandic: Hún finnur seðil í vasanum sínum.

It’s the 3rd-person singular present indicative of the verb finna (to find). The stem finn- takes the ending -ur for hann/hún/það. Present-tense conjugation goes: ég finn, þú finnur, hann/hún/það finnur…

Why is seðill written as seðil here?

Because seðill is a masculine noun whose accusative singular drops one l, yielding seðil. The accusative singular is used because seðil is the direct object of finnur.

Why is vasaNUM in the dative case, and what do its parts signify?

The preposition í meaning “in” (location) governs the dative case in Icelandic. vasi (pocket) in the definite dative singular takes -n (the definite article) plus -um (dative ending), giving vasanum.

How do you know when í takes the dative versus the accusative?

If í expresses a static location (“where something is”), it takes the dative. If it expresses movement into something (“where to?”), it takes the accusative.

Why is the possessive pronoun sín used here as sínum, instead of hennar?

sín is the reflexive possessive pronoun referring back to the subject (Hún). It must agree in gender, number, and case with the noun it modifies—here masculine singular dative—so it becomes sínum. Using hennar would be non-reflexive and could imply a different female person’s pocket.

Could you omit the pronoun Hún at the start? Why or why not?

Yes. Icelandic verbs are inflected for person and number, so subject pronouns are often optional. You could say Finnur seðil í vasanum sínum, though including Hún adds clarity or emphasis that “she” is the one finding the note.

How would the sentence change if you used veski (“purse/bag”) instead of vasi (“pocket”)?

veski is a neuter noun. Its definite dative singular is veskinu, and the reflexive pronoun agrees as neuter dative sínu. The sentence becomes:
Hún finnur seðil í veskinu sínu.