What is the smallest type of stream

Well this might depend on the local terminology for your area, but usually the smallest type of stream is known as a "brook"

The smallest type of stream is called rills. Rills run in tiny channels also called rills. Rivulets and runnels run in gullies. Brooks and creeks run in washes or ravines or arroyos or gulches as well as small valleys with other names.

hope this helps you :)