I know that these forms differ in a sentence use case but what about standalone words? For this specific one when I looked it up on Jisho the ず form is the entry with the meaning of "worthless person" while the ない(ぬ)form is only shown as inflection of 役立つ so I thought that the ない form is only used for the meaning of "something being not helpful/useful" and not "worthless person" specifically. Can these words be used interchangeably?


These words are not interchangeable. 「役立たぬ」「役立たない」 cannot replace 「役立たず」 to mean "worthless person".

As you know, the continuative form (連用形) of a verb can function as a noun, and there are quite a few established nouns, eg:

遊ぶ → 遊び
戦う → 戦い
皿を+洗う → 皿+洗い* → 皿洗い "washing-up"
物を+知る → 物+知り* → 物知り "knowledgeable person"
足を+[踏]{ふ}む → 足+踏み* → 足[踏]{ぶ}み "stomping feet", etc.
(* 複合語/compound of [noun]+[noun form of verb])

役立たず also derived from the continuative form of a verb phrase. ず is the continuative form of the classical negative auxiliary ず (predicative form/終止形). (ぬ is its attributive form/連体形.) Some examples of established nouns:

役に+立たず → 役+立たず* → 役立たず
世間を+知らず → 世間+知らず* → 世間知らず
恩を+知らず → 恩+知らず* → 恩知らず
医者が+要らず → 医者+要らず* → 医者いらず
親知らず🦷, [水入]{みずい}らず, 舌足らず, [土踏]{つちふ}まず👣, etc.
(* 複合語/compound of [noun]+[noun form of verb])

So, the ~ず is the 連用形, and that's why 役立たず can function as a noun. Since it's an established noun, it cannot be replaced with 役立たぬ or 役立たない.

Also please note that the noun 役立たず is read やくたず, not やくたず, while 役立たぬ, 役立たない are read as やくたぬ, やくたない as they are the conjugated forms of the established verb 役[立]{だ}つ (This だ is voiced due to rendaku, as you may know). Their pitch accents are also different:

役立たず - [やくたたず]{LHHLL}

役立たぬ- [やくだたぬ]{LHHHL}

役立たない - [やくだたない]{LHHHLL}

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