Is there a correct gender-neutral singular pronoun ("his" vs. "her" vs. "their")?

Singular they enjoys a long history of usage in English and can be used here: "Each student should save their questions until the end."

However, “singular they” also enjoys a long history of criticism. If you are anxious about being criticized (for what is in fact a perfectly grammatical construction) I would advise rewording to avoid having to use a gender-neutral singular third-person pronoun.

Some rewording strategies that can be employed:

• Use a plural noun: Students should save their questions until the end.
• Use the formal one: One should save one's questions until the end.
• Use his or her: Each student should save his or her questions until the end

OED References for “singular” they

Here for the benefit of those who lack access to its paywalled source are the full and complete operative senses from the Oxford English Dictionary. Per the OED the pronoun they has these specific subsenses for the various scenarios under discussion here:

2. In anaphoric reference to a singular noun or pronoun. 🗨

   ^{Use of they to refer to a singular antecedent has sometimes been considered erroneous.}

_{🗨 Dennis Baron • A brief history of singular ‘they’}

_{…But that’s nothing new. The Oxford English Dictionary traces singular they back to 1375, where it appears in the medieval romance William and the Werewolf. Except for the old-style language of that poem, its use of singular they to refer to an unnamed person seems very modern. Here’s the Middle English version: ‘Hastely hiȝed eche … þei neyȝþed so neiȝh… þere william & his worþi lef were liand i-fere.’ In modern English, that’s: ‘Each man hurried… till they drew near… where William and his darling were lying together.’…}

^{[4 September 2018]}

• 2a. With an antecedent that is grammatically singular, but refers collectively to the members of a group, or has universal reference (e.g. each person, everyone, nobody).

  ^{Sometimes, but not always, used to avoid having to specify the gender(s) of the individual(s) being referred to; cf. sense A. 2b.}

  [[citations ranging from 1350–2014 omitted]]

• 2b. With an antecedent referring to an individual generically or indefinitely (e.g. someone, a person, the student), used esp. so as to make a general reference to such an individual without specifying gender. Cf. ʜᴇ pron. 2b.

  ^{In the 21st century, other th– pronouns (and the possessive adjective their) are sometimes used to refer to a named individual, so as to avoid revealing or making an assumption about that person’s gender; cf. sense A. 2c, and quots. 2008 at ᴛʜᴇɪʀ adj. 2b, 2009 at ᴛʜᴇᴍ pron. 4b, 2009 at ᴛʜᴇᴍꜱᴇʟꜰ pron. 2b.}

  [[citations ranging from 1450–2010 omitted]]

• 2c. Used with reference to a person whose sense of personal identity does not correspond to conventional sex and gender distinctions, and who has typically asked to be referred to as they (rather than as he or she).

  [[citations ranging from 2009–2019 omitted]]

^{_{Copyright © 2019 Oxford University Press. All rights reserved.}}

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To round out the answers here, one is a very proper way to encompass both male and female antecedents.

To boldly go where no one has gone before

For what it's worth, in academic writing I exclusively use the singular third person masculine "he", recognized by many as gender-neutral. Many other languages do this without concern: German and Spanish come to mind.

In everyday speech I will unconsciously use singular "they". In my book, this has unquestionably drifted into very common use in all but the most formal situations.

It is common to write, "Each student should save his or her questions until the end" or to vary his and her throughout your text. In speech you will often hear native speakers say, "Each student should save their questions until the end."

There have been a number of efforts to create new pronouns that would be gender neutral. Some other posters here have given examples. But none of these have really caught on. I'll hazard the prediction that none will. It's difficult enough to invent a new word and get people to use it. To invent a new word in a context where people are routinely using an existing word is very difficult. Pronouns are more difficult still as they are used ALL THE TIME. That is, if you invent a new word for, say, a type of fish, someone writing about fish could mention the new word, define it for those unfamiliar with it, and then use it a few times in the following discussion. Someone expecting to learn something new about fish wouldn't be too jarred to learn a new word along with some new facts. But pronouns are used all the time in many contexts -- heck, it almost EVERY context. And they are used over and over. You don't use a word like "it" or "she" just two or three times in a page of text -- you use it dozens of times, often multiple times within one sentence. Using a new pronoun really stands out and is jarring and distracting.

So in real life, the solutions offerred to this problem are:

1. Stick with the traditional use of "he/him/his" as the generic pronouns. If we are talking about a specific person who is female, of course use "she". If we are talking about a person who is a member of a class which is always or usually female -- like mothers or nurses -- use "she". Otherwise use "he". Disadvantages: May sometimes be misleading, implying that a person must be a male when this is not the case. Advantages: Easy to use. Consistent with traditional use of the language. Offends feminists. (Well, some consider that to be a disadvantage.)

2. Use "they" as a singular pronoun. Disadvantages: Often considered to be grammatically incorrect because of improper agreement with antecedent. Can be misleading as to whether one person or many is intended. Advantages: Is widely enough used that it is generally understood.

3. Use pronouns of both genders together, like "he or she" or "him/her". Disadvantages: Quickly becomes awkward, e.g., "After he or she fills in his or her information on the form, he or she should specify whether he or she wants the form returned to him or her ..." In some cases creates ambiguity about whether one or two people are involved, e.g. "Give this to him or her" -- does that mean that the person I give it to may be of either sex, or that there are two people and I must give it to one or the other? Advantages: Clearly indicates that the person can be of either sex. No grammatical inconsistency.

4. Alternate use of genders, either within a "short space" or a "long space". When done within a very short space this can be confusing, like "He should do A and then she should do B." Clearly sounds like two people are involved. But if, say, you are giving two examples of jobs that must be performed, you could use "he" to refer to the person in the first example and "she" in the second. Disadvantages: Requires an extra effort during composition to continually switch genders, possibly re-writing all the pronouns in a paragraph if you move things around in editing. May still be distracting depending on context. Advantages: Avoids all grammar problems while still maintaining a form of gender neutrality.

5. Reword sentences to always use a plural. For example, instead of saying, "Each student should bring his book", say, "The students should bring their books", etc. Disadvantages: May lead to implicit changes in meaning, e.g. from identifying an individual responsibility to a collective responsibility. In some cases it may be impossible without changing the meaning, like try to reword, "He was the only person who volunteered" to use the plural without changing the meaning. Advantages: No problems with grammatical correctness.

Personally I generally use #1 for formal writing (because it retains strict grammatical correctness), and #2 for informal writing and speech.