Lyrics in screenplays are written in italics.
If it is not already clear from the context that the character is signing, it’s helpful to include the parenthetical “(singing)” the first time it comes up, as foreign dialogue and other special-circumstance dialogue is also put in italics.
Some writers choose to put a “/” at the end of each line of lyrics. Some start a new line. Some do both, and some do neither. All of these are acceptable options.
And if you don’t like Courier’s italics, be sure to check out Courier Prime, a free Courier variant with a unique italic typeface.