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It's worth pointing out that, etymologically speaking, the roots of whether are which/either of two. It's inherently a "binary choice" word, so whereas "I don't know whether it be fish or fowl" is fine, "I don't know whether it be fish or fowl or good red herring" isn't really grammatical. Which is not to say people never use that extended form - but it does sometimes attract criticism.

With "unary choice" forms such as "I don't know whether I like it", the alternative ("I don't like it") can invariably be shortened to "or not" - or simply discarded completely, since it's implicit anyway.

Possibly some will say if only one choice is presented, you should use "if" rather than "whether", but skimming through written instances of "Tell me whether" suggests that most people have always been quite relaxed on that point.

TL;DR: "or not" is never required if the alternative is a simple negation of the stated proposition, but an "or" clause is required in, say, "You must choose whether to write novels or poems" (presupposing that writing, for example, software is not an option currently on offer).

The New York Times' stylebook says or not is often redundant.

It is ordinarily omitted when the clause functions as a noun, e.g. it is the object of a verb or preposition, or subject of the sentence.

However, when a whether clause acts as an adverb, or not is needed.

Check this NYT blog post for more details.

Another test, courtesy of Garner’s Modern American Usage: “or not” is necessary when the phrase “whether or not” means “regardless of whether.”

Similarly to tomothymh, I use "whether" alone unless I intend to convey "regardless of whather." In the latter case, I think "whether or not" is generally preferable to "regardless of whether."

Thus:

Whether I wear a coat depends on the temperature.

But:

I'm going to wear my new red coat whether or not it's cold.

When whether or not is implicit, you may use "or not" for the style or to underline ; it's up to you.

But if, "or not" is not implicit - for ex. in a choice with options, we don't use "or not" : "Whether the car will be black, red or white, I shall drive it."

I'm surprised nobody mentioned this that I saw, so I'll post an answer.

The word whether should be used by itself in the situations you mentioned above. The “or not” is a mistaken crossover from the correct usage of “whether or not” mentioned below. It is often used that way, but when writing it's best to avoid that unnecessary bit.

The phrase whether or not is a condition, used in statements to show that something will or will not happen, regardless of certain other variables:

I'm going to go on strike whether or not anyone joins me!

This would be less correct, however, if used like this:

I don't care whether or not anyone joins me, I'm going on strike!

It depends. Quotes from Times’s stylebook (explained here):

Often "or not" is redundant after whether, but not always. The
phrase may ordinarily be omitted in these cases:

• When the whether clause is the object of a verb: She wonders whether
the teacher will attend. (The clause is the object of wonders.)

• When the clause is the object of a preposition: The teacher will
base his decision on whether the car has been repaired. (The clause is
the object of on.)

• When the clause is the subject of the sentence: Whether the car will
be ready depends on the mechanic. (The clause is the subject of
depends.)

But when a whether clause modifies a verb, or not is needed: They will
play tomorrow whether or not it rains. (The clause modifies play.)

Although perhaps implied in the answers above, I think it could be clarified a bit: Whether is binary. In a sentence where two options are offered, no 'or not' is required - in fact it would be quite cumbersome. However, if the option is to do or not to do, is or isn't, then the 'or not' is required.

I don't know whether I should have the fish or the chicken.
I don't know whether I should eat or not.