Is it safe to boil water in a microwave?

There is one very different issue to be kept in mind - water in a microwave can overheat and "explode" once it is disturbed.

Another poster had exactly this problem a short while ago: Water exploded in Microwave

So follow the usual precautions, e.g. putting a wooden toothpick or a small, very clean stone (chemists have them in their labs) in your vessel. In a pinch, a spoon will do, but not all microwaves handle metal objects well.

If you are worried that heating your water in a microwave might not be sufficient to kill all "nasties", remember that killing bacteria is a function of time and temperature, so you might feel safer if you not only bring your water to a boil, but continue boiling it for another minute or so. I would assume that this is mostly for your psychological benefit, but it certainly won't hurt. The temperature reached is identical for different heating devices as physics dictates the boiling point of water and it can't exceed that as long as it's liquid.

There is no difference. Whether you boil water in a kettle or in a microwave, it reaches a temperature of 100 °C/212 °F at sea level. Not only that, but no method that doesn't involve pressure will get the water to reach a temperature of over 100 °C/212 °F. Water boiled in a microwave is just as safe as water boiled in a kettle.

Even If i put on the tin foil beanie and assert that microwaves are dangerous to food, the microwave remains an ideal way to boil and heat water. Leaving aside the steam dangers which are mostly common to steam regardless of heat source (steam is dangerous) the most dangerous chemical reactions possible from high energy ionization is to produce mild base (HO ions), mild acid (H ions), Hydrogen gas, Oxygen gas, and most of the base will be neutralized by the acid forming water, and the hydrogen and oxygen are not enough to measure. This is actually the safest use for a microwave. It is even slightly more effective at killing bacteria, as some bacteria are also sensitive to microwave radiation as well as heat, but for good practice I would still heat to the recommended temperatures.

Boiling water will kill any live bacteria or viruses that might be in your water. The only thing that can "survive" are bacterial spores. Spores are like seeds or eggs which can hatch to live bacteria. Interestingly, this process is what is used to make salt risen bread...a type of bread leavened by hydrogen-producing bacteria instead of carbon dioxide producing yeast.

The only way to kill spores is the use of a pressure cooker. As another answer mentioned, this does nothing to the chemical dangers, just the biological dangers. If the water is so filthy that you expect there to be actual toxins in the water, purification by RO or distillation is the best bet.

The most significant danger is the super-heated water phenomena. the microwave heats water so gently, that it can actually remain liquid above 100 degrees C. However, the addition of coffee grounds will provide places for steam bubbles to begin to form. The water will then immediately and violently begin to boil, and you could receive some serious burns. If you place a small pinch of coffee into the water while it is in the microwave, it will promote boiling and prevent superheating.

From a bacterial standpoint, boiling water in your microwave is probably sufficient to kill everything of concern.

The time needed to disinfect water depends on the temperature: for example, according to Water Disinfection for International and Wilderness Travelers, five minutes at 60°C, three minutes at 65°C, or one minute at 70°C is sufficient to kill E. Coli. Normally, simply heating the water to boiling and letting it cool is sufficient, but if you've got an exceptionally powerful heat source or unusually cool air, the water may not spend long enough at high temperature. Because of this, the same source recommends holding the water at a boil for one minute as a safety measure.