1. A Pronoun should agree with its Antecedent in person, number and gender; e.g.

i) All students must bring their report cards tomorrow.
ii) Every customer must make his payment by cheque.
iii) Each of the girls has given her papers.

INCORRECT: I am not one of those who believe everything I hear.
CORRECT: I am not one of those who believe everything they hear.

2. When the sex is not known we use the pronoun of the masculine gender for ANYBODY, EVERYBODY, EVERYONE, ANYONE, EACH, etc.; e.g.

i) Anyone can do this sum if he tries.
ii) Everyone has to do his work.
iii) Each of them has submitted his report.

3. When more than two persons or things are spoken we use ANYONE (not EITHER); e.g.

INCORRECT: He is more laborious than either of her four brothers.
CORRECT: He is more laborious than anyone of her four brothers.

4. EACH, EVERY, EITHER, and NEITHER, ANYONE take singular verbs and singular pronouns; e.g.

INCORRECT: Each of the students have done well.
CORRECT: Each of the students has done well.

INCORRECT: Neither of the accusations are true.
CORRECT: Neither of the accusations is true.

INCORRECT: Neither of our two sons has done their homework.
CORRECT: Neither of our two sons has done his homework.

INCORRECT: Each one of us is doing our duty well.
CORRECT: Each one of us doing his duty well.

5. When two subjects are joined by the following, all types of pronouns agree with the first subject

INCORRECT: Suhani as well her sisters has done their exam well.
CORRECT: Suhani as well her sisters has done her exam well.

6. When two subjects are joined by ‘Either – or’, ‘Neither – nor, ‘Not only – but also’, all types of pronouns agree with the nearest subject of the second part of these conjunctions; e.g. 

INCORRECT: Not only the captain but also the teammates were blamed for the defeat of his team.
CORRECT: Not only the captain but also the teammates were blamed for the defeat of their team.

INCORRECT: Neither the teachers nor the Principal is doing their duty with complete dedication.
CORRECT: Neither the teachers nor the Principal is doing his duty with complete dedication.

7. Two singular nouns joined by AND take a singular verb and singular pronouns if only the first of them is preceded by an article, and they take a plural verb and plural pronouns if both of them are preceded by an article separately; e.g. 

INCORRECT: The manager and secretary are present in their room.
CORRECT: The manager and secretary is present in his room.

INCORRECT: The manager and the secretary is present in his room.
CORRECT: The manager and the secretary are present in their room.

8. If two singular nouns are joined by AND, and one or both of them are preceded by EACH or EVERY or NO, take a singular verb, and pronouns and possessives for the subjects are also singular; e.g.

INCORRECT: Each member and each worker has given their consent.
CORRECT: Each member and each worker has given his consent.

NOTE-I: When nouns of EACH and EVERY are of different genders, we use pronouns and possessives of the masculine gender, not of the feminine gender; e.g.

EVERY BOY and EVERY GIRL was given his bag back.

NOTE-II: If antecedent, means the noun preceding EACH is plural, we use plural verb and plural pronouns; e.g.

a) The people each have their own problem.
b) The children each have a special task allotted to them.

NOTE-III: If a sentence has a relative pronoun, in the relative clause we use pronouns for EACH in the plural; e.g.

INCORRECT: Each girl was given a bunch of flowers which pleased her very much.
CORRECT: Each girl was given a bunch of flowers which pleased them very much.

[Here WHICH is a relative pronoun and THEM pronoun for EACH GIRL]

9. If there is use of the pronoun ONE, it should be used throughout; e.g.

INCORRECT: One cannot be too careful about what he says.
CORRECT: One cannot be too careful about what one says.

INCORRECT: One should do his duty.
CORRECT: One should do one’s duty.

INCORRECT: One should do his duty himself.
CORRECT: One should do one’s duty oneself.

10. When ‘MANY A’ is the part of the subject, we use a singular verb and singular pronouns; e.g.

Many a student has not brought his bag today.

11. Collective nouns such as crew, committee, government, army, staff, team, herd, jury take either singular verbs/pronouns or plural verbs/pronouns but when the group is thought of a single unit we use verbs/pronouns in the singular, and when individual members of the group are thought of, the verbs/pronouns are in the plural; e.g. 

INCORRECT: The team is divided in their opinion on the combination.
CORRECT: The team is divided in its opinion on the combination.

INCORRECT: The team are divided in its opinion on the combination.
CORRECT: The team are divided in their opinion on the combination.

12. WE should not use THE SAME in place of a pronoun (HIM, HER, IT, THEM, THEY, etc.), the appropriate pronoun should be used instead of THE SAME here; e.g.

INCORRECT: I gave her a pen, and he gave the same to Sheela.
CORRECT: I gave him a pen, and he gave it to Sheela.

INCORRECT: I think these issues are already settled, and therefore any further discussion on the same is useless.
CORRECT: I think these issues are already settled, and therefore any further discussion on them is useless.