When to sing(from a Summoner's perspective):
Ballad-
-Between pulls.
-Healer is below 25% mana
-Summoner is below ~15% mana.
-Before bosses
-After a SMN or WHM/SCH is rezzed, and ESPECIALLY battle rezzed.

Paeon-
-Between pulls if you're AoEing
-After someone is battle rezzed
-When they ask for it

Requiem-
-If you don't need to sing, and nobody is asking for it, use this. It doesn't drop your own damage, and besides PLD/WAR/MNK everyone uses some kind of elemental damage.