I am wondering how to do this in a novel. Do you write the song (lyrics) in its entirety in one single spurt, or do you spurt a verse at a time interspersed with descriptions of the surroundings/actions/emotions? I really want to write a scene in my novel where the female character sings, but I have no idea how to do this properly. By the way, I am not asking how to format the lyrics.
It depends on the purpose of the song
Songs in fiction can serve multiple purposes. They can provide lore and background details, they can be used as a metaphor, they can portray emotion or conflict, they can foreshadow, reflect or mirror events of the story, they can be used for character development and a whole host of other things. How you need to write them will depend on the message you are trying to convey.
The question you need to ask yourself: Is the message in the content of the song or the characters reaction to them?
The answer to this question should tell your whether you need to include the entire transcript of the song or merely describe its effects on the characters.
Methods I have seen
Include the entire transcript followed by an explanation.
This is the method that takes up the most space in your writing. It should establish the song as an important message or turning point of the story. If you are familiar with Tolkien's work then you will be familiar with this style. Though I would suggest not copying his example of including entire transcript of songs in fictional languages.
Only include a description of the song
This method is good for when the content of the song is unimportant or secondary to the characters reaction. Describe in broad terms the topic and style of the song but keep the narrative focus on how it effects the characters.
You can also use this method if you are struggling to write a song with the desired effect. It is a lot harder to write a good sad song then to say the song was good and made the character feel sad.
Include snippets of the song
This is my personal favourite approach and is a combination of the previous two. Include short snippets of the song, usually the first and last verse and any relevant ones in between. Break up the snippets with exposition on the characters reaction, descriptions of the song and details of its performance.
Using this method is the most flexible and lets you only focus on the part of the song that matter. Additionally it prevents the reader from getting bored skipping the song as I have been known to do upon occasion.
Include the full text in the appendix
This is an option if you are particularly keen on writing the song itself and want to share it, but don't like the way it fits in the main body of the text. There are no drawbacks to this approach but I question how much value it adds since few readers will bother with it.
Examples in fiction
I have already mentioned Tolkien. He used songs and poetry extensively through the Lord of the Rings and the Hobbit. As I mentioned he favoured including entire transcripts of the songs, even the ones in elvish. I wouldn't suggest it but being familiar with the style is a good idea.
The Kingkiller Chronicles (The Name of the Wind) by Patrick Rothfuss heavily focus on songs and their meaning as a central motivation for the main character. He favours the third approach and does it brilliantly. I cannot suggest these books highly enough in general but particularly if you are looking for good examples of songs in fiction.
The Inheritance Cycle by Christopher Paolini is another example. Though guilty of poor, lazy or immature writing (depending on opinion) in a lot of ways, these books do include songs in a good way. He varies his approach for different songs and scenes throughout the series.
I love well used songs in fiction, particularly when used to combine lore with an allegory for character development. I encourage you to write the song and experiment with how to include it. Write the scene multiple times and see which works best. Don't be afraid to modify the song to fit better with the prose around it.
I would say that quite a few different methods are possible. I have read books where a few lines are given, followed by paragraphs or even pages of description of how the characters react, followed by a few more lines, and so on. I have read examples where the entire lyric is given in one place. And ones in which a character describes how he or she reacted to the song, without ever quoting one word. All of these worked, for me. It depends on the effect desired. It also depends on whether you are going to have the character sing an existing song that can be simply referred to by name, or one that you write yourself.
You might look at the various scenes where music is played (not sung) in the Aubry / Maturin books by Patrick O'Brian. There the music, and the shared creation of it, is a significant bond between the characters. It is also used as a way of showing the changing moods of the participating characters. And that is just one example that comes to mind.