When can you omit "that"?

There are a number of instances in English where it is possible and even desirable to omit that.

• that as conjunction with reporting verbs (like learn, discover, find out, know, feel, etc.): I discovered (that) Julian had borrowed my car without my permission.

• after the more common reporting verbs, (e.g. say, tell) it is also entirely natural to omit that in informal speech: I told him (that) I'd be back by ten o'clock but he said he needed me here by nine.

• after certain verbs (e.g. reply, shout) that cannot be omitted and it is not normally dropped after nouns: The Dean of the Humanities Faculty informed the students that the drama dept was going to close.