I am a little confused about the usage of pronouns. I often see people using "their" with words that seem to be singular, for example, "somebody" and "anybody", which looks weird to me. (I.e., one person has recently said to me: "If somebody is hating on someone due to the pitch of their voice...") I would rather say: "...to the pitch of his/her voice...". So who is correct?
In the recent past, male pronouns were used as generic pronouns in English, and the sentence would have typically read "hating someone due to the pitch of his voice."
To modern ears, the older convention sounds dated, sexist and inaccurate, but no one has really come up with a good alternative. The most common practice, as you've noted, is using plural pronouns, which are gender neutral but grammatically incorrect. It is gender neutral and singular, but dehumanizing. Some people use his/her, or more properly his or her, but this is clumsy and wordy. Others switch back and forth between using him and her as the generic pronoun, but you see that more often in writing than in casual speech. You also run the risk of making it sound as if you're pandering to stereotypes --or creating new ones --if you choose poorly which sentence to assign to which gender.
Sometimes you can put the sentence in the second person (as in this sentence) but that makes it sound like you are addressing the person directly. One can also use the word one as a generic pronoun, but it sounds stilted and is not very flexible. The final alternative is to recast the entire sentence in the plural, which fixes the grammar, but at the cost of changing the sentence.
There are also newly coined gender neutral singular pronouns such as zhe but none of those has really caught on yet.