Step 1: Understanding acidity.
The acidity of compounds depends on the ability to donate a proton (H$^+$). In aqueous media, electron-withdrawing groups increase acidity, while electron-donating groups decrease acidity. The presence of an electronegative substituent like F in fluoroacetic acid makes it more acidic than acetic acid or phenol. Alcohols, like methanol (CH$_3$OH), are the least acidic.
Step 2: Conclusion.
The increasing order of acidity is CH$_3$OH (IV) > acetic acid (II) > phenol (I) > fluoroacetic acid (III), corresponding to option (D).