Á töflunni eru bæði dæmi í eintölu og dæmi í fleirtölu.

This is very normal Icelandic word order.

Icelandic main clauses usually follow the V2 rule: the finite verb comes in the second position. If you put Á töflunni first for emphasis or to set the scene, the verb eru must come next:

• Á töflunni = first element
• eru = second element
• bæði dæmi í eintölu og dæmi í fleirtölu = the rest

So the sentence structure is basically:

• On the board
  • are
    • both singular examples and plural examples

This is especially common when introducing what is present somewhere.

Icelandic often does not use a separate dummy word like English there in existential sentences.

English says:

• There are examples on the board.

Icelandic can simply say:

• Á töflunni eru dæmi.

So the idea of there are is expressed just by the structure of the sentence, especially with a location phrase like Á töflunni and the verb eru.

Töflunni is the dative singular definite form of tafla.

Here is the breakdown:

• tafla = board / table / chart
• töflu = dative singular indefinite
• töflunni = dative singular definite = the board

The preposition á can take different cases:

• á + dative for location: on something
• á + accusative for motion onto something

So:

• á töflunni = on the board / on the blackboard
• á töfluna = onto the board

Because the sentence describes where the examples already are, Icelandic uses the dative.

Because the subject is understood as plural.

The sentence mentions two coordinated groups:

• dæmi í eintölu
• og dæmi í fleirtölu

Together, that is plural, so the verb is plural too:

• eru = are
• er = is

Even though the subject comes after the verb, the verb still agrees with it in number.

Bæði ... og ... means both ... and ....

So here:

• bæði dæmi í eintölu og dæmi í fleirtölu
• = both examples in the singular and examples in the plural

This is a very common Icelandic pairing:

• bæði X og Y = both X and Y

You can think of bæði as introducing the first part of a pair, and og linking it to the second part.

That is a very common point of confusion.

Dæmi is a neuter noun, and its indefinite nominative/accusative singular and plural have the same form:

• dæmi = example
• dæmi = examples

So the form itself does not tell you singular vs. plural here. The context does.

In this sentence, both instances are understood as plural in meaning:

• dæmi í eintölu = examples in the singular
• dæmi í fleirtölu = examples in the plural

This kind of same-form singular/plural pattern is fairly common with some neuter nouns in Icelandic.

Yes, Icelandic could sometimes leave it out if the meaning is clear, but repeating it makes the structure more explicit and balanced.

So:

• bæði dæmi í eintölu og dæmi í fleirtölu

clearly presents two parallel items.

Repeating the noun is especially natural in teaching or explanatory language, where clarity matters. It helps the reader immediately see that the sentence is talking about two kinds of examples.

These are standard grammatical expressions:

• í eintölu = in the singular
• í fleirtölu = in the plural

They use the preposition í, which often means in.

The nouns are:

• eintala = singular
• fleirtala = plural

In grammatical descriptions, Icelandic very often uses these fixed expressions with í:

• orðið er í eintölu = the word is in the singular
• nafnorðið er í fleirtölu = the noun is in the plural

So this sentence is using normal grammar terminology.

Because the sentence is talking about things in a general, non-specific way.

• dæmi means examples, not the examples
• í eintölu and í fleirtölu are fixed grammatical expressions, and they normally appear without the definite article

So the sentence is not referring to one special set called the singular or the plural. It is just classifying the examples by grammatical number.

That is why the sentence uses:

• dæmi
• í eintölu
• í fleirtölu

rather than definite forms.