
sonderlich
particularly
Mostly used in negative or interrogative sentences and means 'especially' or 'to a high degree'. It softens a statement.
Das Wetter war heute nicht sonderlich gut.
The weather wasn't particularly good today.
particularly / peculiar
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particularly
Mostly used in negative or interrogative sentences and means 'especially' or 'to a high degree'. It softens a statement.
Das Wetter war heute nicht sonderlich gut.
The weather wasn't particularly good today.

peculiar
Describing a person or thing as strange, odd, or unusual. This meaning is rather rare today.
Er war schon immer ein etwas sonderlicher Mensch.
He had always been a somewhat peculiar person.
'Sonderlich' (particularly/peculiar) is derived from 'Sonder-', which means 'separate' or 'special'. It is related to words like 'besonders' (especially/special) or 'absondern' (to separate), which explains the meaning 'special' or 'different from normal'.
Er hat seine eigenen, sonderlichen Ansichten zu diesem Thema.
He has his own peculiar views on this topic.
In the meaning 'particularly', 'sonderlich' is used almost exclusively in negative sentences. The combination 'nicht sonderlich' means 'not particularly' or 'not very'.
Ich war von dem Film nicht sonderlich begeistert.
I wasn't particularly enthusiastic about the film.
The meaning 'strange' or 'peculiar' is rather rare today and can sound old-fashioned or literary. It describes a person or their behavior as odd.
Der alte Mann im Nachbarhaus war ein sonderlicher Kauz.
The old man in the neighboring house was a peculiar character.
especially / special
Besonders can express a high degree in affirmative, negative, and interrogative sentences. Sonderlich has a similar meaning here, but it is usually used in negations, qualifications, or questions and softens the statement. For positive emphasis, besonders is therefore the better choice.

Das Essen in diesem Restaurant ist nicht sonderlich gut.
The food at this restaurant is not particularly good.
Das Essen in diesem Restaurant ist besonders gut.
The food at this restaurant is especially good.