What is the correct use of 'Many a'?

The second sentence is correct.

Many a girl was appearing in the examination.

"Many a" is always followed by a singular noun/ pronoun/ verb.

Many a girl or the second option is correct. Many a always uses a singular noun and is more formal than the word many. It is not much common and is used usually in newspapers and famous writings. Many a is used to indicate a large number of something.

If you want to say many a x, you need to use a singular noun. So many a girl would be correct. However, if you want, you could say many girls appeared / are appearing instead. if you take out the "a / an", many becomes plural and you can then use plural nouns like "girls" or "cars".

Many a/an...

The quick answer to the question is "Many a girl" is correct, and "Many a girls" is incorrect.

The fixed expression many a/an... is more formal than the single word many, and it is much less common. Many a/an... is used mainly in literary writing and newspapers. Like the adjective and pronoun many, many a/an... is used to indicate a large number of something. However, it takes a singular noun, which can be followed by a singular verb.

Here are some examples:

1. It remained a mystery for many a year. [=for many years]
2. I've been there many a time. [=many times]
3. Many a politician has promised to make changes. [=many politicians have]

Source: Merrian-Webster Learner's Dictionary.