Неприятности начались вчера.

Yes. Неприятности is the plural form of неприятность, which means trouble, problem, or unpleasantness.

Russian often uses the plural here in a very natural way, just like English can say troubles began or problems started. So неприятности начались means that a series of problems or unpleasant events began.

Singular would be:

Неприятность началась вчера.
= A problem / an unpleasant incident started yesterday.

Because the subject, неприятности, is plural.

In Russian past tense, verbs agree with the subject in number, and in the singular they also agree in gender.

So:

• неприятность началась = singular feminine
• неприятности начались = plural

That is why the verb ends in -лись here.

The dictionary form is начаться.

This verb means to begin, to start, or more literally to get started.
It is the perfective partner of начинаться.

So:

• начаться = to begin / to start up (perfective)
• начинаться = to be beginning / to start (imperfective)

The form начались is:

• past tense
• plural
• from начаться

The -ся ending marks a reflexive verb, but in many cases it is not truly reflexive in the English sense.

Here, начаться means to begin in the sense that something begins by itself:

• Фильм начался. = The film started.
• Урок начался. = The lesson began.
• Неприятности начались. = The troubles began.

Compare:

• начать = to begin something / to start something
• начаться = to begin / to start (itself)

For example:

• Он начал разговор. = He started the conversation.
• Разговор начался. = The conversation started.

So the -сь here is part of the normal verb form.

This is about aspect.

Начались is from the perfective verb начаться. Perfective is used when the speaker sees the action as a complete event or a single starting point.

So Неприятности начались вчера means something like:

• the troubles began yesterday
• yesterday was the point when they started

If you used начинались instead, that would be imperfective, which would sound different and would usually need a special context. It could suggest a process, repetition, or background description.

In this sentence, the speaker is pointing to a definite moment of beginning, so начались is the natural choice.

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

You can tell this from the meaning and from the verb agreement:

• неприятности = the things that began
• начались = plural verb agreeing with that subject

So this is not an object; it is the subject.

Вчера is an adverb, meaning yesterday.

It does not change for case, gender, or number. It stays вчера.

It tells you when the action happened:

• начались вчера = started yesterday

Russian adverbs are often very convenient because they do not decline.

Yes. Russian word order is more flexible than English.

The neutral order here is:

Неприятности начались вчера.

But you could also say:

• Вчера начались неприятности.
• Начались неприятности вчера.

These versions are grammatically possible, but they shift emphasis a bit.

For example:

• Неприятности начались вчера.
  Neutral: the troubles started yesterday.

• Вчера начались неприятности.
  Stronger focus on yesterday.

Russian often uses word order to manage emphasis, topic, and new information.

Russian has no articles.

So неприятности can mean:

• the troubles
• troubles
• some troubles

The exact interpretation depends on context.

That is completely normal in Russian. Learners often want to insert something like the, but Russian simply does not use articles.

The singular noun неприятность is feminine.

A useful clue is that many nouns ending in -ость are feminine:

• новость = news item
• скорость = speed
• возможность = possibility
• неприятность = trouble / unpleasantness

In the singular past tense, that feminine gender shows up on the verb:

• Неприятность началась.

In the plural, gender is no longer shown in the verb:

• Неприятности начались.

Yes, it is a common and useful word.

It can refer to:

• problems
• troubles
• unpleasant situations
• difficulties caused by bad events

It is slightly broader than a single concrete problem. Depending on context, it can sound like trouble in a general sense.

Examples:

• У него неприятности на работе.
  He is having trouble at work.

• Из-за этого могут быть неприятности.
  This could cause problems.

So in your sentence, it suggests that some unpleasant developments or problems started yesterday.

The usual stress is:

• неприя́тности
• начали́сь
• вчера́

Stress matters in Russian pronunciation, so it is worth learning words together with their stress.

A rough pronunciation guide:

• неприя́тности = ne-pri-YAT-nos-ti
• начали́сь = na-cha-LEES
• вчера́ = fchee-RA

The exact pronunciation will depend on your accent, but the stressed syllables are the important part to remember.