У этой кроссовки грязная подошва.

Russian usually expresses possession with the pattern:

у + possessor (genitive) + possessed thing

So У этой кроссовки грязная подошва is literally something like:

At this sneaker, [there is] a dirty sole.

That is the normal Russian way to say this sneaker has a dirty sole.

English uses have, but Russian often does not.

Because у requires the genitive case.

The base form is эта кроссовка = this sneaker.
After у, it changes to:

• эта → этой
• кроссовка → кроссовки

So:

• эта кроссовка = nominative
• у этой кроссовки = genitive after у

Because this is the genitive singular of кроссовка.

Forms:

• nominative singular: кроссовка
• genitive singular: кроссовки

This can be confusing because кроссовки is also the nominative plural form meaning sneakers.

So кроссовки can mean either:

• of the sneaker (genitive singular), or
• sneakers (nominative/accusative plural)

In this sentence, it is clearly singular genitive, because it follows этой, which is feminine singular genitive.

Yes. Кроссовка is the singular form: a sneaker.

However, in real life people often talk about sneakers as a pair, so the plural кроссовки is very common. That is why learners often notice the plural first.

This sentence is about one sneaker, so the singular makes sense.

Because подошва is the thing being talked about as the main subject-like element of the sentence.

In this structure:

У этой кроссовки = possessor
грязная подошва = what it has

So подошва stays in the nominative, and the adjective agrees with it:

• подошва — feminine singular nominative
• грязная — feminine singular nominative

That is why it is грязная подошва, not грязной подошвы.

Because Russian often omits есть in the present tense, especially when talking about someone's features, parts, or characteristics.

Compare:

• У него голубые глаза. = He has blue eyes.
• У этой кроссовки грязная подошва. = This sneaker has a dirty sole.

Adding есть here would usually sound unnecessary, and in this kind of sentence it is normally left out.

Because it agrees with подошва, and подошва is a feminine noun.

Agreement in Russian means the adjective matches the noun in:

• gender
• number
• case

So:

• подошва = feminine, singular, nominative
• грязная = feminine, singular, nominative

It is about one shoe.

That is shown by этой кроссовки, which refers to a single feminine noun кроссовка.

If you wanted to talk about the pair, you would normally use plural forms, for example:

У этих кроссовок грязные подошвы.
= These sneakers have dirty soles.

So singular sentence = one sneaker
plural sentence = the pair / multiple sneakers

Yes. Russian word order is flexible, although the original order is natural.

У этой кроссовки грязная подошва is a neutral way to present the information.

You could also hear:

У этой кроссовки подошва грязная.

That version puts more focus on грязная as the important new information, almost like:

As for this sneaker, the sole is dirty.

So the meaning stays close, but the emphasis changes.

You know because of the word этой.

Этой is:

• feminine
• singular
• genitive

So it can only go with a singular feminine noun here: кроссовки = genitive singular of кроссовка.

If the noun were plural, you would expect этих:

• у этих кроссовок = of these sneakers / these sneakers have ...

So этой кроссовки must mean of this sneaker, not these sneakers.