Microwave ovens are safe according to the FDA.
More Info: The FDA is the regulating body for the sale of all microwave ovens and has set forth safety guidelines that manufacturers must meet. A microwave that is in perfect working order and is used according to manufacturer’s instructions is safe.
Do Microwaves Leak Radiation?
When questioning safety, most consumers want to know how much radiation they are being exposed to. The FDA sets a standard to limit the amount of radiation emission from a microwave in its lifetime to no more than 5 milliwatts of radiation per square centimeter, an amount that the FDA considers safe for humans. Most microwaves tested by the agency had little to no radiation leakage even at the lowest levels. There are testing kits available to ensure that there isn’t any radiation leaking from your microwave. 
Did you know that microwave exposure decrease dramatically the further away you are from the source?
Unsafe Plastic Containers
Another question about microwave safety is the use of plastic containers. Speculation abounds about plastic containers that are used in the microwave leaking chemicals that then migrate into the food. This is another area that the FDA regulates. For the FDA to give a plastic container its microwave safe stamp of approval, the product is thoroughly tested. In order for these products to be reliable, they must be used according to directions. For example, if a container suggests it is only reliable after one use, it should be thrown away.
The real concern is the use of plastic containers that are not intended for use in the microwave such as margarine tubs or whipped cream containers. Since the FDA does not test these types of plastic containers, and there is a possibility that they can melt, they should not be used in the microwave. 
It is also unsafe to defrost meat in its original packaging if it sits on a foam tray. These are not labeled for microwave use and can potentially melt leaching chemicals into your meat. Remove the meat from the packing and defrost on a microwave safe container. 
When Are Microwave Ovens not Safe
Damaged microwaves do pose a safety risk. Radiation may leak from a microwave that has a damaged seal, broken door latch, or broken door hinges. Do not use a microwave that is not in perfect working order. 
Used Improperly microwaves can pose a safety hazard. For example, microwaving items that do not belong in the microwave such as clothing can be a fire hazard.
Home canning requires that the foods to be preserved are first brought to a temperature that will kill all harmful bacteria during processing. The FDA advises against the use of microwave ovens for this purpose as it is believed that they cannot produce or maintain the required temperatures. 
Overheating water past its boiling point can cause what is known as erupted hot water phenomena. Water that is overheated without any additions such as coffee grounds can heat the water past its boiling point while appearing as if it hasn’t boiled. The water then erupts out of the cup when disturbed and can cause serious burns. 
Cooking with Microwaves
 Harvard Health Publications
Microwaving Food in Plastic: Dangerous or Not?
 Food Safety and Inspection Service-USDA
Microwave Ovens and Food Safety
 Child Welfare League of America