Learn Hardcore Japanese: tukue no ue ni megane ga arimasu.

In 机の上に眼鏡があります。, we are not saying “Glasses are desk-top,” we are stating the existence and location of the glasses: “There are glasses on the desk.”
So Japanese uses ある/あります, not です, for this kind of sentence.

What is the difference between ある and いる, and why do we use あります with 眼鏡?

Japanese separates existence verbs by whether something is animate or not:

いる / います – for living, animate beings (people, animals, sometimes robots, characters)
- 犬がいます。 – There is a dog.
ある / あります – for inanimate things (objects, places, plants, events, ideas)
- 本があります。 – There is a book.

眼鏡 (glasses) is an inanimate object, so you must use ある → polite form あります.
眼鏡がいます would sound wrong, as if the glasses were a living creature.

What exactly does の do in 机の上?

の links two nouns and often corresponds to “of” or “’s” in English.

机 – desk
上 – top / above / surface
机の上 – literally “the top of the desk” → “on top of the desk”

So the structure is:

Noun A + の + Noun B → “Noun B of Noun A”
机 + の + 上 → “the top of the desk”

Why do we need に after 上 in 机の上に? Does に mean “on”?

に is the basic location particle used with ある / いる to mark where something exists.

The pattern is:

Place + に + Thing + が + ある / いる

In this sentence:

机の上 – “the top of the desk”
机の上に – “on (the top of) the desk” as a location where something exists

Strictly speaking:

上 gives the idea “top / surface / above”
に marks that as the position of the object

So に itself doesn’t specifically mean “on”; it just marks location.
“On,” “in,” “under,” etc., come from the combination of に with words like:

上（うえ）に – on / above
中（なか）に – in / inside
下（した）に – under / below

Could I say 机に眼鏡があります。 instead? Is that wrong?

It’s not strictly ungrammatical, but it sounds unnatural for the meaning “on the desk.”

机の上に眼鏡があります。 – There are glasses on the desk.
(Clearly on the surface.)
机に眼鏡があります。 – “There are glasses at / on the desk.”
This is vague and unusual; a native speaker would almost always say 机の上に to express “on the desk (surface).”

Japanese normally uses a relational noun like 上, 中, 下, 前, 後ろ with に to be clear about position:

机の中に本があります。 – There is a book in the desk.
机の下にカバンがあります。 – There is a bag under the desk.

So for “on the desk,” stick with 机の上に.

What is the role of が after 眼鏡, and why not use は here?

が is marking 眼鏡 as the thing that exists — the grammatical subject of あります.

In sentences of the form:

Place に Thing があります / います。

が usually marks something that is:

new information
being presented as “there is / there are ...”

So in 机の上に眼鏡があります, the sentence is naturally heard as:

“On the desk, there are glasses.” (introducing that fact)

は is a topic marker and has a different nuance.
Using は here would change what is being highlighted (see next question).

How would the meaning change if I said 眼鏡は机の上にあります。?

Both are grammatical, but the focus changes.

机の上に眼鏡があります。
- “On the desk, there are glasses.”
- Typical answer to: “What is on the desk?”
- Focus: introducing the existence of glasses.
眼鏡は机の上にあります。
- “As for the glasses, (they) are on the desk.”
- Typical answer to: “Where are the glasses?”
- 眼鏡 is now the topic (something already known), and the new information is the location.

So:

が → often used when introducing what exists.
は → used when the item is already in the conversation and you’re saying something about it (here: where it is).

Is the word order fixed, or can I say 眼鏡が机の上にあります。 instead?

The word order is not strictly fixed as long as particles stay with the right words and the verb is at the end.

Both are natural:

机の上に眼鏡があります。
眼鏡が机の上にあります。

Subtle nuance:

Starting with 机の上に lightly sets “on the desk” as the setting:
- “On the desk, there are glasses.”
Starting with 眼鏡が lightly sets “glasses” as the starting point:
- “The glasses are on the desk.”

In everyday conversation, both versions are fine. Just keep あります at the end.

How do I say this sentence in casual speech?

The casual (plain) form of あります is ある.

So:

Polite: 机の上に眼鏡があります。
Casual: 机の上に眼鏡がある。

In very casual speech, が is often dropped:

机の上に眼鏡ある。

For learners, it’s safest to keep が; just remember that ある (no ます) is the casual verb form.

How do I turn this into a yes–no question like “Are the glasses on the desk?”?

In polite Japanese, add か to the end:

机の上に眼鏡がありますか。
“Are the glasses on the desk?” / “Is there a pair of glasses on the desk?”

In casual speech, you usually just raise your intonation at the end (and you might also use the casual form):

机の上に眼鏡ある？ – “Are the glasses on the desk?”

Does 眼鏡 mean “a pair of glasses” or “glasses” in general? How do I show singular or plural?

Japanese generally does not mark singular vs plural the way English does.

眼鏡 by itself can mean:
- “glasses,” “a pair of glasses,” or “pairs of glasses”
  depending on context.

To be explicit about number, Japanese uses numbers + counters, for example (using a generic counter):

眼鏡が一つあります。 – There is one pair of glasses.
眼鏡が二つあります。 – There are two pairs of glasses.

In real life, people often rely on context and don’t say the number unless it matters. Just remember that 眼鏡 doesn’t change form for plural.

Can any particles be dropped in natural spoken Japanese in this sentence?

Yes, in casual conversation some particles are often omitted, especially が.

Possible forms:

Full, polite:
机の上に眼鏡があります。
Casual, keep all particles:
机の上に眼鏡がある。
Casual, drop が (very common):
机の上に眼鏡ある。

に is less often dropped here, because it clearly marks the location with ある.
For learners, it’s best to keep all particles until you’re very comfortable with them.