I haven't read all of the other responses, so I might be repeating their answers. But, I want to comment on a couple of things. First of all, if he's going to sing and play until he falls asleep, then that's his mechanism to put himself to sleep, and he's probably going to do it whether you tire him out or not! It's a blessing when a child finds their own way to fall asleep, and some children just need that wind-down time. Think of what it takes you to fall asleep at night - we all have some trick we use to help us fall asleep, even if it's just to consciously clear our minds. So, I wouldn't be disturbed by his need to sing and play for a while. (My son does it, and is still able to put himself to sleep in a room full of other kids without being disruptive. They learn social rules!)
Secondly, there is no strict rule as to when you need to move your son to a toddler bed. If he's climbing the rails, you would want a safer bed. But, if he's still sleeping comfortably, keep him there! The earliest I moved a child was at 22 mos because we had a second child coming. A friend of mine didn't move her son until he was three because it just worked well for them since he didn't go to bed well. Please don't feel pressured to move him just because he hits a certain age. (My son is almost 22 mos, and we are expecting to move soon. For that reason, it will probably be after his 2nd birthday before he's moved simply to get him through the transition without too many upsets in his routine!)
Hope that helps!