hängenbleiben
to get caught
To get a body part or item of clothing snagged on something and not be able to get free immediately.
Ich bin mit meinem Ärmel am Nagel in der Wand hängengeblieben.
I got my sleeve caught on the nail in the wall.

to get caught / stick
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to get caught
To get a body part or item of clothing snagged on something and not be able to get free immediately.
Ich bin mit meinem Ärmel am Nagel in der Wand hängengeblieben.
I got my sleeve caught on the nail in the wall.

'Hängenbleiben' is a separable verb. In a main clause, 'hängen' comes at the end. In a subordinate clause, the verb stays together, and in the infinitive, 'zu' comes between the two parts.
Beim Klettern blieb er an einem Ast hängen.
While climbing, he got caught on a branch.
The perfect tense of 'hängenbleiben' is always formed with the auxiliary verb 'sein' (to be). This applies to all meanings of the verb.
Mir ist von dem Gespräch nur ein Satz hängengeblieben.
Only one sentence from the conversation has stuck with me.
The meanings 'to repeat a school year' and 'to linger at a place' are almost exclusively part of colloquial language. They are rarely used in formal or written contexts.
Nach dem Konzert sind wir noch in einer Bar hängengeblieben.
After the concert, we ended up staying in a bar.
In the physical meaning, the preposition 'an' with the dative is almost always used to describe what someone got caught on.
Sie ist mit ihrem Schal am Fahrradlenker hängengeblieben.
She got her scarf caught on the bicycle handlebar.
When something remains in memory, the person who remembers is the dative object. The thing that is remembered is the subject of the sentence.
Diese Melodie ist mir den ganzen Tag hängengeblieben.
This melody stuck with me all day.
to stay / remain
'bleiben' and 'hängenbleiben' overlap when something remains in a place or in memory. 'bleiben' is neutral and general: someone or something is still there, or a state does not change. 'hängenbleiben' is more specific and often colloquial. It can mean that you stay somewhere longer than planned, or that something sticks in your memory. Learners may confuse the words because both can express 'not going away'. 'bleiben' fits normal and planned situations; 'hängenbleiben' fits when something is accidental, unexpected, or especially noticeable.

Nach der Feier sind wir noch lange bei Freunden hängen geblieben.
After the party, we ended up staying at our friends' place for a long time.
Jonas ist beim Termin geblieben, obwohl der Zug Verspätung hatte.
Jonas kept the appointment, even though the train was delayed.