1.Neither John nor Mary has done their homework.
2.Neither John nor Mary has done his or her homework.
Which pronoun is suitable, singular or plural?
1.Neither John nor Mary has done their homework.
2.Neither John nor Mary has done his or her homework.
Which pronoun is suitable, singular or plural?
To begin with, the second option is not possible here. The formula "his or her" is reserved for generic subjects, when you don't know or intend to specify the sex.
That person who just left forgot his or her coat.
A teacher should respect his or her students.
So is the first one okay? More or less. In most settings, you would use "their" in a case like this.
In higher registers (maybe as high as an academic paper), you would avoid "their" in this sentence. The solution would depend on the context. For example, if you're talking about the same class and the same assignment, you would probably say "the":
Neither John nor Mary has done the homework.
But if you couldn't do that, you might have to reword entirely:
John and Mary have not done their homework.
That said, be aware that the correct use of "Neither ... nor" itself suggests a higher register. Hence, it doesn't jive very well with this use of "their". So you might reword it in a casual setting too.
One way of capturing the emphasis that "Neither ... nor" gives in a casual setting might be:
John and Mary both haven't done their homework!