I have already stored used peanut oil in the refrigerator for one week. I had strained it through coffee filters, and it had looked very clean and clear. Now it looks cloudy and milky. Is it OK to use? I had used it to cook fish and hush puppies. 
aswine is correct.  The cloudiness is a function of the amount of saturated fats in the oil.  The more saturated fat an oil contains, the higher the temperature it solidifies.  As Joe commented, olive oil will become cloudy/solidify in the fridge as well.      This is the reason manufacturers 'hydrogenate' oils to produce margarines.  The more hydrogen atoms they can attach to an oil the more 'saturated' it becomes and the more solid it is at room temperature.  Lard, bacon grease, highly saturated, solid at room temperature.  Some oils are less saturated so they stay liquid at room temperature and the fridge.  Your peanut oil is in the middle range, so it just got cloudy in the fridge.  As far as microbes growing in the oil, they definitely can.  Exposure to air (oxygen) and oil breakdown (exacerbated by heat) are bigger culprits.  This breakdown has a few indicators, the primary one for me being a rancid smell.  Keeping oils sealed from air and stored in the fridge/freezer will help slow down the breakdown.  Keep the oils tightly sealed, away from light and cool and refrigeration is generally not required.  Oils I don't use often I buy in smaller quantities so I wind up using them up before they go rancid sitting on the counter.  Some oils, like walnut go rancid quickly and should be stored in the fridge.  Experience will guide you here.  I personally don't worry too much about microbes in my oils.  They usually start to smell bad before they are a health concern.  I'm also not a doctor and don't cook for anyone with impaired immune systems, so you need to make your own calls here.  In the South, bacon grease is often left out at room temperature.  This can be done due to its high salt content.  Just make sure not to mix in other greases or oils.    Finally, (and this will likely start a fight) butter can be left out at room temperature (for a while) safely.  This is great if you like spreading soft button on your toast.  The length of time it can be left out is a function of the temperature of the room and the amount of salt in your butter.  Some people use butter bells for this function.  Butter will not last as long on the counter as in the fridge.  Also, your nose will tell you when you've left it out too long and it needs to be discarded.  (We eat it too quickly at my house for that to happen though.  :-) 