Vasaklúturinn er í vasanum mínum.

They are:

• vasaklútur = handkerchief
• vasi = pocket
• minn = my / mine

The sentence uses inflected forms:

• vasaklúturinn = vasaklútur
  • the definite ending
• vasanum = dative singular definite of vasi
• mínum = dative masculine singular of minn

Because Icelandic usually adds the to the end of the noun instead of using a separate word.

So:

• vasaklútur = handkerchief
• vasaklúturinn = the handkerchief

Here the noun is masculine singular nominative, so the definite form is vasaklúturinn.

Er is the present tense of vera, the verb to be.

So in this sentence it simply means is.

• ég er = I am
• þú ert = you are
• hann/hún/það er = he/she/it is

Here the subject is singular, so er is the correct form.

Because í can take different cases depending on meaning.

In this sentence, í means in in the sense of location: the handkerchief is already inside the pocket. With that meaning, í takes the dative.

So:

• dictionary form: vasi
• dative singular: vasa
• definite dative singular: vasanum

That is why the sentence has í vasanum = in the pocket.

Because Icelandic possessives must agree with the noun they describe in:

• gender
• number
• case

Here mínum describes vasanum.

And vasanum is:

• masculine
• singular
• dative

So minn changes to mínum.

Compare:

• vasaklúturinn minn = my handkerchief
  nominative masculine singular
• í vasanum mínum = in my pocket
  dative masculine singular

It refers to the pocket.

So the structure is:

• í vasanum mínum = in my pocket

It does not mean my handkerchief. If you wanted that, you would say:

• vasaklúturinn minn = my handkerchief

A useful clue is that mínum is placed right after vasanum and agrees with it grammatically.

That is a very common Icelandic pattern.

A neutral, everyday way to say my pocket, my book, my car, etc. is:

• vasinn minn
• bókin mín
• bíllinn minn

So noun + definite article + possessive is extremely normal in Icelandic.

English usually puts the possessive before the noun, but Icelandic very often puts it after.

Because Icelandic normally keeps the definite article in this pattern.

So:

• vasanum mínum is the ordinary way to say my pocket
• literally, it looks something like the pocket my

To an English speaker this may seem redundant, but in Icelandic it is completely natural.

Yes, you can, but it usually sounds more emphatic or contrastive.

• í vasanum mínum = the neutral, usual way to say in my pocket
• í mínum vasa = more like in my pocket (as opposed to someone else’s)

So both are possible, but the version in your sentence is the more neutral everyday phrasing.

This is an important Icelandic pattern.

With location (in), í takes the dative:

• í vasanum mínum = in my pocket

With motion into (into), í takes the accusative:

• í vasann minn = into my pocket

So the case helps show whether something is already somewhere or moving there.

The word order is the normal declarative pattern:

• Vasaklúturinn = subject
• er = verb
• í vasanum mínum = prepositional phrase

So it is:

Subject + Verb + Location

This is very natural Icelandic word order. Because Icelandic is a verb-second language, if you move another element to the front, the verb still stays in second position. For example:

• Í vasanum mínum er vasaklúturinn.

That still means the same thing, but with a different emphasis.