How many times has a divorce actually occurred, and is the wife permissible to her husband after the repetition of these formulas: "If I wake up from sleep and find you here, then you are divorced," and "You are divorced" (during a quarrel), and "You are divorced three times," and "If I come home and find you in it, then you are divorced"?

Your statement to your wife, "You are divorced," effects a single, explicit divorce. As for your statement, "You are thrice divorced," uttered at once, the majority of jurists hold that it counts as three divorces, and your wife becomes irrevocably separated from you with a major divorce (baynunah kubra), meaning she is not lawful for you until she marries another man and he legitimately consummates the marriage with her. However, Shaykh al-Islam Ibn Taymiyyah is of the opinion that uttering "thrice divorced" at once counts as a single divorce. As for your statements, "If I wake up and find you here, I will divorce you," and "If I come home and find you, I will divorce you," these are merely promises of divorce and do not constitute a divorce. It is advised to refer to the Sharia (definition) courts.