"Do's" and "don'ts" or "do's" and "don't's"?

DOs & DON'Ts

that would be appropriate.

Do's and Don't's

seems correct but because of the plethora of apostrophes in the latter word, I prefer

Do's and Don'ts.

You can see a nice discussion about this subject on this blog post (When apostrophes go bad: Do's and Don'ts).

"Things to DO and NOT DO".

"Do's and Don'ts" seems to be preferred, but let's be honest it's not an elegant construction either way.

Although "Do's" and "Don'ts" seem to be the popular usage, it would be correct to write "Dos" and "Don'ts" for a couple of reasons:

• It is a plural form of the word you are trying to express, not possessive;
• It is not a contraction either — the contraction "do's" would mean "do is";
• "Don'ts" is the correct plural form of "don't" so "dos" would be the correct plural form of "do".

I personally don’t like subjective arguments based on how an expression seems or feels. Statistics is better, but it doesn't eliminate the case of millions of flies that can't be wrong.

There are several style guides for written communication which I think should be consulted first and foremost, and there appears to be two documented options:

do’s/don’ts

From The elements of typographic style, section 5.4.3 Omit the apostrophe from numerical plurals [1, p. 88] (emphasis mine):

In the interests of typographic hygiene, unnecessary hyphens should likewise be omitted. […] Apostrophes are needed for some plurals, but not for others, and inconsistency is better than a profusion of unnecessary marks. Thus: do’s and don’ts; the ayes have it but the I’s don’t; the ewes are coming but the you’s are staying home.

For what it is worth, Apple Style Guide (accessed 2021-02-26) also suggests do’s/don’ts format and uses it in the guidelines for UI design and for brand and photography.

dos/don’ts

From The Chicago Manual of Style [2] (emphasis mine):

7.14 Plurals of noun coinages. Words and hyphenated phrases that are not nouns but are used as nouns usually form the plural by adding s or es. (If in doubt, consult an unabridged dictionary like Webster’s Third New Inter­national, which indicates the preferred inflected forms for most nouns, including all of the examples below.)

ifs and buts
dos and don’ts
threes and fours
thank-yous
maybes
yeses and nos

[…]

7.30 Contractions. In contractions, an apostrophe normally replaces omitted letters. Some contractions, such as won’t or ain’t, are formed irregularly. Colloquialisms such as gonna or wanna take no apostrophe (there being no obvious place for one). Webster’s lists many common contractions, along with alternative spellings and, where appropriate, plurals. Note that an apostrophe—the equivalent of a right single quotation mark (’ not ‘)—is always used to form a contraction (see 6.117).

singin’   gov’t   ’tis (not ’tis)   dos and don’ts   rock ’n’ roll

P.S. Note that the majority of style guides underline that a typesetter's apostrophe (’) must be used and not a “lazy” typewriter apostrophe (').

References

1. Bringhurst, R. The Elements of Typographic Style, 3rd ed.; Hartley & Marks: Point Roberts, WA, 2004. ISBN 978-0-88179-205-8.
2. The Chicago Manual of Style, 17th ed.; University of Chicago Press: Chicago; London, 2017. ISBN 978-0-226-28705-8.