Go to the bomber guarding the entrance to the observatory wearing a mask.  He will comment on it.

When the thief, Sakon, steals the bag from the old lady, hookshot the bag.  Sakon will explode!

Wear the Mask of Truth and pick up the dog in Clock Town.  He'll comment on how much he hates Deku Link!  Or turn into a Zora.  The dog will sit and look at you funny.

At various points, Tatl will insult you.  On Ikana, for instance, try talking to the family after healing the father.  Or pay attention when you first talk to Kafei.  Navi may have been annoying, but she was much nicer.

Look into the sky at night.  Look around.  Eventually, you may see Orion.  The Big Dipper's there as well.

(Submitted by Black Moon Eclipse):   On day three after midnight, you'll find the Swords Teacher at Clock Town hidden behind a wall. He's blubbering like a baby!