die Sache
thing
An unspecified object or thing.
Was ist das für eine komische Sache auf dem Tisch?
What is that strange thing on the table?

thing / matter
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thing
An unspecified object or thing.
Was ist das für eine komische Sache auf dem Tisch?
What is that strange thing on the table?

The meaning of 'Sache' (thing/matter) often changes with number. In the singular ('die Sache'), it usually means a single object or a matter. In the plural ('die Sachen'), it very often refers to personal belongings or luggage.
Das ist eine private Sache, über die ich nicht sprechen möchte.
That is a private matter that I don't want to talk about.
To distinguish between 'Gegenstand' (object) and 'Angelegenheit' (matter), pay attention to the context. If it is about a concrete, touchable object, it is a thing. If a problem or topic is being discussed, it is a matter.
Nimm diese klebrige Sache vom Sofa weg!
Take that sticky thing off the sofa!
When 'Sache' (matter) describes a matter or issue, it is often combined with adjectives that express an evaluation. Typical examples are: 'eine klare Sache' (a clear-cut matter), 'eine schwierige Sache' (a difficult matter), 'eine ernste Sache' (a serious matter), or 'eine tolle Sache' (a great thing).
Der Sieg der Mannschaft war am Ende eine klare Sache.
In the end, the team's victory was a clear-cut matter.
The idiom 'zur Sache kommen' (to get to the point) is a fixed unit. It is used to ask someone to say the most important thing and stop digressing.
Wir haben nicht viel Zeit, also lasst uns direkt zur Sache kommen.
We don't have much time, so let's get straight to the point.
The meaning 'Fall' (case) is found almost exclusively in legal or very formal contexts. Signal words here are 'Gericht' (court), 'Anwalt' (lawyer), 'Akte' (file), or 'Verfahren' (proceedings).
Der Anwalt prüft die Dokumente in dieser Sache sehr genau.
The lawyer is examining the documents in this matter very carefully.
In the expression 'seine Sache gut machen' (to do a good job), the possessive pronoun is adapted to the person. So you also say 'ich mache meine Sache gut' (I do a good job) or 'ihr macht eure Sache gut' (you all do a good job).
Der neue Mitarbeiter ist sehr motiviert und macht seine Sache wirklich gut.
The new employee is very motivated and is really doing a good job.
The phrase 'nicht jemandes Sache sein' (not to be someone's thing) is a fixed expression. The possessive pronoun ('meine', 'deine', 'seine', etc.) is adapted to the person you are talking about.
Ich helfe dir gern beim Kochen, aber Abwaschen ist nicht so meine Sache.
I'm happy to help you with cooking, but doing the dishes isn't really my thing.
case / fall
'der Fall' and 'die Sache' can both mean a matter or a problem. However, 'der Fall' sounds more specific and often means a particular situation, a concrete occurrence, or something that is being officially investigated. 'die Sache' is more general and more everyday; it can mean a topic, a problem, or a matter. Learners may confuse the words because both refer to something that needs to be discussed or clarified. When it is about one clearly defined incident or process, 'Fall' is more precise. When speaking generally about a topic or problem, 'Sache' sounds more natural.

Wir besprechen die Sache morgen in Ruhe.
We'll discuss the matter calmly tomorrow.
Bei der Arbeit hat der Kunde den Fall noch einmal genau erklärt.
At work, the customer explained the case again in detail.