What it says on the tin.
What do I do when I'm out of deck? I can't seem to find a rule.
You reshuffle the deck once you have to draw or discard the top of an empty deck.
(For clarity:) But not if you have to reveal more cards than are in the deck. 
I.e., Revealing doesn't trigger shuffling the trash into the deck, only, as Ronway stated, when you have to draw or discard and there's nothing there to draw or discard.
So you won’t reshuffle the villain deck if they are out of cards?
That is correct that you would not reshuffle the villain deck if they run out of cards. Though when a card from the villain deck needs to be drawn to be put into play or discard the top card of it than you would reshuffle it.
Thanks!