I hate it when my girlfriend sings.
What about her singing bothers you?
She can't even sing properly.
Her singing might not be good, but you should let minor annoyances go and not focus on them. We all have things about us that are annoying, but you should still respect your girlfriend.
I don't care. I would be happier in a new relationship.
If you're going to break up with your girlfriend over something as small as you don't like her singing, you are going to struggle to find someone that makes you happy. Every potential partner is going to do things that you find annoying. You need to learn to be able to accept someone's flaws if you want to be with someone long term.