There are four types of birds in the game. The blue bird, the blue rain bird, the yellow bird and the yellow rain bird. Blue birds drop silver stars, yellow birds drop gold stars. Yellow birds however only appear to drop gold stars when you are out of ammo. They don’t just normally appear in the game. They are sort of an emergency system to help you out in a rough spot.
The blue bird will appear in a normal game about once every 15 to 20 seconds (or about that). However neither the yellow or the blue bird will fly in the rain. If you are playing a level where it is raining you will need to have purchased the blue and yellow rain birds from the shop portion of the game. The rain bird doesn’t come by as often as a normal blue bird does. They will fly in the rain but will only appear about half as often as a normal blue bird would in a non rain game.