На рубашке появилась такая заметная складка, что мне пришлось достать утюг.

На рубашке means on the shirt — on its surface.

That fits складка, because a wrinkle/crease is thought of as something on the fabric.

• на рубашке = on the shirt
• в рубашке = in the shirt / inside the shirt

So in this sentence, на is the natural choice.

Also, рубашке is in the prepositional case because it follows на in a location meaning:

• рубашка → на рубашке

Появилась is the past tense, feminine singular, perfective form of появиться.

It agrees with складка, which is feminine:

• складка появилась

Why perfective? Because the sentence talks about a wrinkle appearing as a completed event:

• a wrinkle showed up / formed

If you used the imperfective появлялась, it would sound more like:

• was appearing
• used to appear
• kept appearing

That is not the idea here. Here, one noticeable wrinkle appeared, and that led to a result.

Такая заметная складка means such a noticeable wrinkle.

Both adjectives agree with складка:

• feminine
• singular
• nominative

So:

• такая
• заметная
• складка

The word такой / такая / такое / такие often means:

• such
• or, in this pattern, so ... that

Here it helps set up the result clause:

• такая заметная складка, что...
• such a noticeable wrinkle that...

This is a very common Russian pattern meaning so/such ... that.

Structure:

• такой / такая / такое / такие + noun, что + result

In this sentence:

• такая заметная складка, что мне пришлось достать утюг
• such a noticeable wrinkle that I had to get out the iron

Compare:

• Он был такой уставший, что сразу уснул.
  = He was so tired that he fell asleep immediately.
• Был такой шум, что я не мог работать.
  = There was such noise that I couldn’t work.

So что here introduces the consequence/result.

Because складка is the subject of the verb появилась.

What appeared?

• складка

So it stays in the nominative:

• складка появилась

The phrase на рубашке is just a location:

• On the shirt, a wrinkle appeared

In this sentence, складка means something like:

• wrinkle
• crease
• fold

For clothing, crease or wrinkle is the most natural English idea.

A few nuances:

• складка can mean a fold in fabric
• on clothes, it often refers to an unwanted crease/wrinkle
• context tells you whether it is neutral or undesirable

Here it is clearly undesirable, because it was bad enough that the speaker had to get the iron.

Because появилась emphasizes that the wrinkle showed up / formed.

Compare:

• На рубашке была складка.
  = There was a wrinkle on the shirt.
  This simply states its existence.

• На рубашке появилась складка.
  = A wrinkle appeared on the shirt.
  This highlights that it came into being.

So появилась makes the situation feel more dynamic and gives a clearer cause for what follows.

Мне пришлось is a very common way to say I had to.

It comes from прийтись / приходиться, which often works in an impersonal construction:

• мне пришлось + infinitive
• literally something like it fell to me to...
• naturally: I had to...

So:

• мне пришлось достать утюг
• I had to get out the iron

Compare:

• я должен был достать утюг
  = I was supposed to / was obliged to get the iron
  This can sound more like duty, obligation, or expectation.

• мне пришлось достать утюг
  = I had to get the iron
  This sounds more like the situation forced it.

That is why пришлось fits very well here.

Because пришлось uses the person affected in the dative.

Pattern:

• мне пришлось
• тебе пришлось
• ему пришлось
• нам пришлось, etc.

Examples:

• Мне пришлось ждать. = I had to wait.
• Нам пришлось уйти. = We had to leave.

So the Russian structure is not literally I had to, but more like:

• to me, it became necessary to...

That is why you get мне, not я.

Because this is an impersonal past-tense construction.

With impersonal пришлось, Russian uses the past form in neuter singular:

• пришлось

It does not agree with мне.

So even though the meaning is I had to, the verb form stays the same:

• мне пришлось
• тебе пришлось
• ему пришлось
• ей пришлось
• нам пришлось

This is normal for impersonal expressions in Russian.

Here достать means to get out, take out, or fetch.

So:

• достать утюг = get out the iron / take the iron out

This suggests the iron was put away somewhere, and the speaker had to go get it.

Why not just взять?

• взять утюг = take/pick up the iron
• достать утюг = get the iron out from wherever it was stored

So достать is a bit more specific and fits the situation well.

It is also perfective, which suits a one-time completed action.

Because утюг is the direct object of достать:

• достать что? → утюг

So it is in the accusative case.

For inanimate masculine singular nouns, the accusative usually looks the same as the nominative:

• утюг → утюг

Compare:

• nominative: утюг
• accusative: утюг

If it were animate, the form would often change:

• я вижу брата
  accusative of брат

But with утюг, there is no visible change.

The word order is natural, but not the only possibility.

На рубашке появилась такая заметная складка, что мне пришлось достать утюг.

This order does a few useful things:

1. На рубашке sets the scene first
2. появилась складка introduces the new event
3. такая заметная ..., что... builds toward the consequence

Russian word order is flexible, but changing it changes emphasis.

For example:

• Такая заметная складка появилась на рубашке, что мне пришлось достать утюг.

This puts stronger emphasis on such a noticeable wrinkle.

The original version sounds very natural and balanced.

Not necessarily — but it strongly suggests that.

Literally, the sentence says:

• the wrinkle was so noticeable that I had to get out the iron

So the explicitly stated action is only:

• достать утюг = get out the iron

In real life, that strongly implies the next step would be ironing the shirt, but Russian does not directly say that here.

Yes. Заметный / заметная / заметное / заметные can mean:

• noticeable
• visible
• obvious
• quite prominent

So такая заметная складка could be understood as:

• such a noticeable wrinkle
• such an obvious crease
• such a visible fold

In this sentence, noticeable is probably the safest general translation, but the sense is that the wrinkle was hard to ignore.

A simple breakdown is:

• На рубашке = on the shirt
• появилась = appeared
• такая заметная складка = such a noticeable wrinkle
• что = that
• мне пришлось = I had to
• достать утюг = get out the iron

So the structure is:

• location
• main event
• such ... that result pattern
• impersonal had to construction
• infinitive phrase

That makes it a very useful sentence for seeing several common Russian patterns working together.