My rule no. 1 - "If it hurts, stop" - applies here. If your throat is so sore it hurts to speak, let alone sing, for goodness' sake don't! Pain is always an indicator that something is wrong. Shut up and let your poor voice heal.
However, lesser ills are not necessarily a reason to stop singing. If it doesn't hurt to sing - even if your throat is a bit sore - it's often ok to do so. But there is a second consideration.
Sometimes voices that are below par simply don't want to work. They don't hurt, but trying to sing is like wading through treacle, and the voice is a shadow of its normal self. If this is how you feel, singing is pointless. You will only wear yourself out and you may do damage. So even though there is no pain as such, rule no. 1 still applies. Stop, and let your voice heal.
But what if there's no pain and your voice is working ok, but every time you try to sing you get the Frog Chorus? This often happens at the end of a cold, when there's still a lot of catarrh lying around. My general advice would be to let the frogs sing. Don't try to avoid them by singing "above" them or pushing through them - that only makes the catarrh worse. And remember my rule number 3 - you can't hear your own voice. You hear the frogs. We generally don't - but we do notice if you compromise your vocal production in order to avoid them.
Above all though, if you have to sing with a voice that isn't at its best, you MUST maintain good vocal technique. So many people start singing too quietly when they have a cold, because they think that preserves their voice. It doesn't. Many people also stop breathing deeply and supporting their voices because they are worried that the sound may crack. Reducing the breath and support is the WORST thing you can do to a voice that is not well. If the sound cracks when you support it properly, you shouldn't be singing.
So if you have a cold or a cough, sing normally or not at all. And let the frogs sing too.
