I cannot take a shower at night.
“des Nachts” cannot possibly be right, since Nacht is feminine and so in the genitive case both the singular and the plural definite article would be “der”. And in any case wouldn’t the more natural way to say this be “Ich kann nachts nicht duschen”?
It is correct but hopelessly obsolete/archaic/poetic.
You’re right, simply “nachts” is what people would say in modern german.
Edit 1: That said, “eines Nachts” is still in use, just like “eines Tages”!
Edit 2: You can still find occurrences, especially if someone deliberately wants to sound archaic. For example in the Shrek movie there’s Fiona’s curse: “Des Nachts ist es so, bei Tage ganz anders. So ist es eben… bis zum ersten Kuss der wahren Liebe… und dann… ihrer wahren Gestalt!”
Just wondering: does that mean Nacht was once masculine/neuter? Or do cases overall work differently in archaic German?
A quick research revealed the claim that people came up with it by analogy to the other time specifiers, which are all masculine:
Die Form “nachts” ist durch Analogie über die Formen “morgens, mittags, abends” entstanden, wonach dann auch “des Nachts” in Anschluß an “des Tages, des Morgens, des Mittags, des Abends” entstanden ist.
But it’s pretty much the only evidence I found.
Nothing against Shrek but by “des Nachts” another quote comes to my mind:
“Es sind die schlechtesten Deutschen nicht, die unruhig werden, wenn sie des Nachts an Deutschland denken.”
(Willy Brandt, in reminescence to Heine’s famous “Nachtgedanken”)