In the beginning of "The Butts Family", Rarity accuses Applejack of having stolen her tree. She denies it...even if the tree has the word "Rarity" carved in it...and she left a note saying "Ha Ha! I stole your tree! Signed AJ!"
Also Applejack is apparently a kleptomaniac.
For the same episode, Applejack's cousin thinks Ponyville is racism-free, Applejack tries nervously to confirm this only for Twilight to snicker in the background.