Под раковиной появилось влажное пятно.

Because под here means under in the sense of location, and with that meaning it normally takes the instrumental case.

• раковина = base dictionary form, nominative
• раковиной = instrumental singular

So:

• под раковиной = under the sink / beneath the sink

This is a very common Russian pattern:

• под столом = under the table
• под кроватью = under the bed
• под деревом = under the tree

Some Russian prepositions can take different cases depending on the meaning. Под is one of them.

With location (where?), под takes the instrumental:

• под раковиной = under the sink

With movement toward a position under something (to where?), под takes the accusative:

• под раковину = to under the sink / under the sink as a destination

Compare:

• Кот сидит под столом. = The cat is sitting under the table.
• Кот залез под стол. = The cat climbed under the table.

In your sentence, nothing is moving under the sink; the wet spot is located there, so instrumental is used.

Because it agrees with пятно, which is a neuter singular noun.

• пятно = neuter singular
• past tense verbs in Russian agree in gender and number with the subject

So:

• masculine: появился
• feminine: появилась
• neuter: появилось
• plural: появились

Since пятно is the thing that appeared, the verb must be появилось.

Example:

• появился человек = a person appeared
• появилась проблема = a problem appeared
• появилось пятно = a spot appeared

Russian word order is much more flexible than English word order. The subject does not have to come first.

In this sentence:

• Под раковиной появилось влажное пятно.

the order sounds natural because it presents the information like this:

1. Where? — under the sink
2. What happened? — appeared
3. What appeared? — a wet spot

This kind of order is very common when introducing something new into the scene.

A version with the subject first is also possible:

• Влажное пятно появилось под раковиной.

That is grammatical, but it shifts the emphasis a little. The original sentence sounds more like noticing a new development in that location.

Появилось means appeared, showed up, or came into existence. It suggests that the wet spot was not there before, and now it is.

So:

• Под раковиной было влажное пятно.
  = There was a wet spot under the sink.
  This simply states its existence.

• Под раковиной появилось влажное пятно.
  = A wet spot appeared under the sink.
  This emphasizes the new appearance of the spot.

So появилось gives a stronger sense of change or discovery.

Влажное is an adjective, and it must agree with пятно in:

• gender: neuter
• number: singular
• case: nominative

Since пятно is neuter singular nominative, the adjective also takes the neuter singular nominative ending:

• masculine: влажный
• feminine: влажная
• neuter: влажное
• plural: влажные

So:

• влажное пятно = wet spot / damp patch

This is standard adjective-noun agreement in Russian.

It is nominative, because it is the subject of the sentence.

Even though it comes after the verb, it is still the thing that appeared.

A useful clue is the verb agreement:

• появилось matches пятно as a neuter singular subject

So the structure is essentially:

• [Under the sink] [appeared] [a wet spot].

Russian often allows the subject to stand after the verb, especially when introducing new information.

Пятно is a fairly broad word. Depending on context, it can mean:

• spot
• stain
• patch
• mark

In this sentence, влажное пятно most naturally suggests:

• a wet spot
• a damp patch

It does not specifically mean a full puddle. If the speaker wanted to emphasize a puddle, they might use a different word such as лужа.

So влажное пятно sounds like some moisture has shown up under the sink, possibly from a leak.

Usually, yes. Раковина commonly means sink or washbasin.

Depending on context, it could be:

• a kitchen sink
• a bathroom sink

Russian does not specify which one here. The sentence just says under the sink.

A learner should also know that раковина can mean a shell in other contexts, but in this sentence под раковиной clearly means under the sink.

Появилось is perfective.

The verb pair is:

• появляться = imperfective
• появиться = perfective

Here, the past tense form появилось comes from появиться.

Why perfective? Because the sentence refers to a completed event: the wet spot came into existence or showed up.

If you used the imperfective in the past, it would usually suggest repeated action, process, or background description, which would be less natural here.

So the perfective fits the idea:

• there wasn’t a wet spot before
• then one appeared

Grammatically, yes, but it would usually mean something different and sound less natural in this context.

Появлялось is imperfective, so it suggests something like:

• it used to appear
• it was appearing
• it would appear from time to time

For example:

• После каждого мытья посуды под раковиной появлялось влажное пятно.
  = After every dishwashing, a wet spot would appear under the sink.

But for a single newly noticed spot, появилось is the natural choice.

A natural stress pattern is:

Под ракОвиной появИлось влАжное пятнО.

Word stresses:

• под
• ракОвиной
• появИлось
• влАжное
• пятнО

Stress is important in Russian because it is not always predictable from spelling.

Yes, very naturally.

Even though the sentence literally just says that a wet spot appeared under the sink, the most likely real-world implication is that:

• something is leaking
• water has gotten under the sink
• there may be a plumbing problem

Russian often leaves that implication unstated, just as English does. The sentence itself is neutral and descriptive, but the context strongly suggests a leak.