So I was womdering:
If I buy a song and have rights to use it in a video, can I sing it? Can I sing it to an instrumental version of the song?
Better look up the copyright laws. I think you can sing all day long, but as for recording..? not sure, but probably not, strictly going by the by the rules.
Should probably look it up for yourself.
Pardon my corny sense of humor
You'll need to contact the copyright holder.
Without getting into detail (details are hundreds of pages), there are different copyrights for a song's lyrics, a song's music (melody), and an individual recording (a cover). If you are recording yourself singing, you're creating a new recording, but are still responsible for licensing the lyrics and melody from the copyright holder.
This might be easier than you think. Strictly for audio recording of a cover, there is a thing called a compulsory mechanical license. Basically if a song is published in the US, the copyright holder is actually required to grant a license at the statutory royalty rate to anybody wanting to record a cover. Again this is just for audio recordings.
Performing a cover live - It's up to the venue to get a license from a Performance Rights Organization like ASCAP that allows live covers and it's safe to assume the venue has one unless the venue states otherwise. If you, as the artist, perform a cover in a venue that doesn't have a license, you don't get in any trouble. The venue on the other hand, is in hot water.
YouTube - A cover on a video service like YouTube requires a synch license. This is the one that can get hairy but there are clearing houses like Easy Song Licensing for obtaining a proper license for most songs. On top of that, YouTube itself has negotiated synch licensing for a lot of songs. You can check what songs are already covered and which ones aren't at the YouTube Music Policies page. If you aren't covered by YouTube or can't get a proper license through a clearing house then it's off to the copyright holders directly to see what can be worked out. Believe it or not, as long as it's a respectful cover a lot of artists will grant a free license to the average Youtuber.
Some excellent blog posts by a copyright attorney"