stinken
to stink
To give off a very strong and unpleasant smell.
Der Müll in der Tonne fängt an zu stinken.
The trash in the bin is starting to stink.

to stink
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to stink
To give off a very strong and unpleasant smell.
Der Müll in der Tonne fängt an zu stinken.
The trash in the bin is starting to stink.

'Stinken' (to stink) is a strong verb. The past forms are 'stank' in the simple past and 'hat gestunken' in the present perfect.
Der Fisch von letzter Woche hat furchtbar gestunken.
The fish from last week stank terribly.
The figurative meaning, where a situation 'stinkt' (stinks) to someone or someone 'vor Geld stinkt' (is filthy rich), belongs to colloquial language and should be avoided in formal contexts.
Mir stinkt diese ganze Bürokratie gewaltig!
All this bureaucracy really stinks!
To name the source of a bad smell, the preposition 'nach' is used followed by the dative. The subject is often the impersonal 'es'.
Hier stinkt es nach altem Käse.
It stinks of old cheese in here.
In the colloquial meaning 'to be annoyed', a fixed structure is used: 'Es stinkt' + person in the dative + subordinate clause with 'dass' or 'zu'.
Ihm stinkt es, dass er nie für seine Arbeit gelobt wird.
It stinks to him that he is never praised for his work.
The expression 'stinken vor' is used almost exclusively in the combination 'vor Geld stinken' (to be filthy rich), to describe extreme wealth in a disparaging or envious way.
Seit er im Lotto gewonnen hat, stinkt er vor Geld.
Since he won the lottery, he has been filthy rich.
to annoy
'nerven' and 'stinken' can both express that you find something unpleasant. 'nerven' directly means that something bothers or irritates you. 'stinken' actually means that something smells bad. Colloquially, however, you can say 'Das stinkt mir' (That stinks to me / I really don't like that) when you find a situation unfair, annoying, or very unpleasant. The difference is important: for a smell, only 'stinken' fits. For a person or thing that is bothering you, 'nerven' is more normal. 'stinken' in the figurative sense sounds more casual and often stronger.

Der Müll im Hausflur stinkt schon seit zwei Tagen.
The trash in the hallway has been stinking for two days now.
Beim Sport hat mich die laute Musik aus der Halle genervt.
During sports, the loud music from the hall annoyed me.