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-ず and -ぬ are two alternatives to the negative form -ない / -ません. But I noticed that depending on the word, it's either -ず or -ぬ, although it seems like some words can take both suffixes. Some examples I have encountered:

# -ず

知らず (lyric in song "タッチ" by Younha)  
わからず (lyric in song "愛するもの" by Angela Aki/アンジェラ・アキ)  
せず (song title: "油断せずに行こう")  
なれず (lyric in song "未来へ" by Kiroro/キロロ)

# -ぬ

気づかぬ (lyric in song "ありがとう..." by KOKIA)  
届かぬ (song title: "届かぬ想い")  
ならぬ (lyric in song "春雷" by YeLLOW Generation)

# both -ず and -ぬ

とらず / とらぬ  
できず / できぬ

So are there rules for choosing between -ず and -ぬ?
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conjugations negation auxiliary-ず
What is the difference between the negative forms -ず and -ぬ?
This is an excellent question. I always think of this and don't know why I
didn't think to post it.

Anyway, **-zu** seems to usually be an adverbial, connective form (often as
**-zu** or **-zu ni** ), whereas **-nu** seems to be a modifier.

>   * 誰も知らずに家に入った - I entered the house without anyone (else) knowing
>   * 誰も知らぬ秘密 - A secret that no one knows.
>

That's just my perception of them up to this point. I actually have a theory
about **-nu** that I really want to research into more.
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-ぬ is an archaic form of -ない. I suspect its use in song lyrics has more to do with fitting the word into the right number of syllables; as far as I know, there is no difference in meaning.

-ず, on the other hand, indicates that one action took place without or in the absence of another action (the one with -ず). For example, 待たずに先に行く: to go on ahead _without waiting_.
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In the modern form, ず is only used as an adverbial (食べずに出る leave without
eating). ぬ can replace ない.

In 文語, the grammar used in writing until the reformations after WWII and still
at least partially in many forms of poetry, songs, and very formal documents,
the use of ず and ぬ was/is grammatically constrained in a manner no longer
present in modern Japanese. ぬ was used with the 連体形 form of a word, meaning
that it connected with a following noun (and it still does that).

ず, on the other hand, connected with the 未然形 and was used for negative
assertions. It had conjugations, too, but those are really really rare now
(ざら、ざり、ざる、ざれ、ざれ).

Just remember this: ず can end a sentence, but ぬ can't; ぬ modifies nouns, but ず
doesn't: 知らぬ人、人知らず.

Of course they've also snuck into idioms and 慣用語: 我知らず (despite
oneself)、暑さ知らずの所 (a place that knows no heat), 知らぬ顔 (pretend
ignorance)、知らぬ存ぜぬで押し通す (persist in asserting one's innocence). (新和英大辞典)
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There is a distinction between 'ず' and 'ぬ' as in the other answers, but for
the combination with the verbs, there is no limitation, despite your
description. I do not understand why you came up with that limitation.

> 知らず / 知らぬ
>
> 分からず / 分からぬ
>
> せず / せぬ
>
> なれず / なれぬ
>
> 気づかず / 気づかぬ
>
> 届かず / 届かぬ
>
> ならず / ならぬ
>
> とらず / とらぬ
>
> できず / できぬ
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> So are there rules for choosing between {-ず} and {-ぬ}?

Well, yes. Just look at how they are used, it should be obvious! It's probably
something as below:

proposition-verbず(に), proposition.  
親に挨拶をせずに、部屋に入ちゃった。  
He rushed into his room without saying hello to his parents.

proposition-verbぬ  
こういう行為を許さぬ！ I won't tolerate such behaviour! (draw out your sword, and slash
your opponent for he danced on the table)

Proposition-verbぬ-nominalgroup…  
言ってはいけぬ言葉は許さない! (draw out your sword, and slash your opponent for he said
"sh*t")

You'll see that it's no more that old-fashioned¹ "〜ないで" and "ない".

¹: I don't like the qualification "archaic" for terms/words/conjugations I
happen to use :P PS: the two last examples may not be very natural…
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I came across the ぬ form in this proverb 触らぬ神に祟りなし which was loosely
correlated with the english "Let sleeping dogs lie" I hear and use the {-ず}
suffix all the time however, so in my limited experience it's more common.
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Both ず and ぬ came from the archaic negator ず in Heian period. The ず had
conjugations ず (未然), ず (連用), ず (終止), ぬ (連体) and ね (仮定), as every Japanese
learns in high school today. The ぬ was originally the 連体形 (noun modifying
form) of ず.

After that, spoken Japanese lost the distinction between 連体形 and 終止形 in almost
all cases. So, now we use ぬ or its variant ん as a sentence terminal form, and
the word is officially called ぬ instead of ず. You might know that ん is heavily
used in Kansai dialects.

The modern ぬ(ん) is conjugated as ぬ/ん, ×, ず, ぬ(ん), ぬ(ん) and ね. The noun
modifying form is still used commonly, and the forms ず and ね are remaining in
some fixed expressions such as 〜ず、 (中止法), 〜ずに (=〜ないで) and 〜ねば (=〜なければ).

The archaic ず and its conjugations are, however, still used in poetry and
proverbs with the classical grammar. 親の心子知らず and 弘法は筆を選ばず are examples.
Anyway, you don't need to use the archaic ず or the modern ぬ when you speak.
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