Using a Zoom H4N to record singing...

The zoom's microphone is not so bad. It will work. What you need to do is experiment with placement. Try a few inches away then 12 inches. You may find you need to rig a pop filters if the zoom is very close.

Be sure and dis-able the automatic gain control. Turn it up as high as it will go but not so high it comes close to clipping.

Next you need to find a room where the microphone sounds good. Some time dead center in bed room work as they have enough soft furniture and being in the center keeps you away from the walls.