I think the primary considerations are convenience (how much effort is it to set up and use the system?) and time spent (how long does the system take to heat the water?).

A standard electric stove can have 2500W elements, and most of this energy will go into a kettle sitting on the element and thus heat the water. Even a big built-in microwave won't be rated at much more than 1000W, and a fair percentage of that is wasted in heating the magnetron (I've found citations of 30% to 50% wasted power). So, all else being equal, a kettle on a stove will heat a body of water 2-4 times as fast as a microwave oven.

If you're instead talking about electric kettles, then in the US (120VAC) they're 1000W to 1750W with very high efficiency, so they'll still be faster than anything but a stove or a commercial microwave. In the lands with 220VAC power, electric kettles can be up to 3000W, which would be even faster than a stove-top kettle.

With a mug of water the convenience of a microwave may outweigh the lost time, but if you're heating a liter or more of water then you probably want to pull out that kettle.

Bear in mind that I'm using an electric kettle, rather than a stove-top one.

First, the advantage of a kettle is that it is quite efficient, and turns itself off once the water is boiling. This as opposed to the microwave, which only stops after a set time, rather than relying on the condition of the water.

Second, a microwave can cause water to superheat, particularly in a ceramic or glass container. This means that there is a certain risk that the water will "jump out" at you once you add something to it or put a spoon in.

Additionally, most kettles have more power than the average microwave, so the kettle is probably faster to boil water.

Convenience

Using an standalone electric kettle device provides for a convenient and safe way to boil water without minding.

• Because of automatic power-off feature, you know the water will be at the correct temperature, not too cool and not too hot.
• No danger of boiling dry in devices with a safety Off feature.

If you are delayed in using the water, just hit the button for a quick re-boil. Some devices can even maintain a desired temperature automatically.

The kettle is more efficient. Nearly 100% of the energy is converted by Joule's law into heat into the water (the kettle is also heated and it makes some noise so it's never 100%, but near). Microwave ovens at the other hand usually have a 60-65% power efficiency just for microwave generation. So if both electrical appliances have a same power, the kettle will be faster and cheaper in energy. Kettles and also cheaper, but microwave ovens can be used for more things that just boiling water. Having a more specific purpose appliance, a more generic one, or both, will be interesting according to your needs, just like bikes, cars, running shoes, tools or whatever you imagine: how often will you use it, can a more generic and less specific solution be enough, what's the price, etc.

Where I live, natural gas is far cheaper than electricity. So, using a kettle on the stove is “best” from this point of view. I happen to have solar power, so in the daytime it’s cheaper to use my own electricity instead.

I used to have another item not discussed on this thread: an instant boiling water tap. That was certainly the handiest, until it broke.

There are a couple reasons why people might heat water on the stove instead of in the microwave - not all of them are strong reasons, but they can influence people nonetheless.

One point is, there is a perception that the method of heating changes the water. I've heard that the flavor is affected, if water is heated by microwave or on stovetop. I've heard that there may be deeper changes in the water - there was an anecdote floating around about how plants watered with previously-microwaved and cooled water vs never-microwaved water did not thrive, implying possible health issues with microwaved water (generally issues assumed to be long term and long scale use of microwaved water, not casual use).

I am not saying there is a difference in flavor or in health benefits, but there is some kind of anecdotal, hand-wavish perception there that people may be drawing from.

On the more practical side, it can be difficult to figure out how much the water needs to heat. On a stove top, the water will simmer then boil in visible markers, in the microwave, it just sits still and can vary in temperature (random example, it took a minute and a half longer to heat my mugful of water in the winter as opposed to summer, in my old microwave. Haven't sussed out my new microwave yet). If you're not familiar with the microwave's quirks (or compensating for the conditions), water can easily come out a little cool, or superheated, which is kind of annoying - especially if you're using it for something where water temp actually matters, like brewing tea.

I suppose part of the difference is the microwave takes less time and is supposed to be more reliable, which is why the small differences between times with different models or conditions seems so much more annoying than stoves, which everyone knows will take more time.

Also on the practical side, it's usually easier to heat larger amounts on the stovetop. An occasional mugful is quick in the microwave, but if you're having several cupfuls in in a row (or for several people) it's easier to keep a big pot warm on the stove than heat several mugs in succession.

As for the difference between a pot on the stove and a kettle - well, you've got me there. I suppose there are benefits to having a pot dedicated to just boiling water (much easier to clean, I guess?), and the kettle is specialized for the task - but I've always just used a when heating water and never missed the difference.

The other answers so far have made it clear that energy efficiency is the main advantage of using an electric kettle. What nobody has mentioned, however, is the fact that most modern kettles have a "minimum fill" line which usually corresponds to several cups' worth of water. I live alone, so if I want a cup of tea I always microwave it, because it's still going to be more efficient than heating twice as much water as I need in a kettle. When people visit, I put the kettle on.

Again in the context of tea and coffee, microwaves have another slight efficiency advantage over what the numbers show, because you don't have to heat the water all the way to boiling. Remember that you can't actually drink boiling water - even after the time it takes to prepare your beverage, the cooldown from adding milk, etc., you may need to let it stand a while before you can drink it. After a bit of practice, a microwave user can hit the exact water temperature for instant enjoyment, and no more. (Plus, I'm convinced the tea tastes better when the water is microwaved - probably because of the slightly lower temperature; pouring water straight from the kettle onto the teabag gives it a weird flavour sometimes.)

I don't know if I've answered your question or the inverse of it, but those are my reasons for favouring the microwave for small volumes of water. (FWIW, I come from a family of mug-microwavers - everybody I know thinks we are weird - and in several decades of experience I've never witnessed the superheating problem that so many people associate with water in the microwave. Just use sensible heating times and it's not at all dangerous.)

An electric kettle is about 80% efficient, for around 1200W of power, and takes around 2 minutes to boil 12oz of water.

An electric stove is about 30% efficient. A 1250W burner would take about 5 minutes to boil 12oz of water; even a 2500W burner (the highest I've seen) would take 2.5 to 3 minutes (you would lose some efficiency as the heat increases).

A microwave oven shows the efficiency directly: a 1100W microwave is not using 1100W but is outputting 1100W (using something higher, probably 1500W for example). A 1100W microwave would take 2.5 or so minutes to boil the water, a 900W microwave would take a bit over 3 minutes.

As such, the electric kettle is clearly the superior choice, but it's not necessarily that much faster for a small amount of water. The microwave may be better when you consider the amount of time it takes to get out the kettle, fill it, etc. compared to directly heating a cup of water.

Numbers largely come from this article.

Automatic Temperature Control

This is the main reason I use an electric kettle instead of any of the other options, and has yet to be mentioned in any of the other (admittedly very good) answers.

I primarily boil water for steeping tea, and as tea aficionados will tell you, many types of tea taste better when steeped with hot but not boiling water.

Traditionally you would either have to wait for the water to reach the desired temperature (say 180 degrees Fahrenheit for example), or boil the water and then wait for it to cool to the desired temperature, either by guessing the temperature or using a thermometer. With my electric kettle I can simply select what temperature I want to produce and my kettle will heat the water until it reaches the desired state. Of course with the microwave you can't leave a thermometer in the water while heating (at least I would be wary of doing so), so you would have to either frequently pull the water out to test it, or boil the water and then wait for it to cool back to the desired temperature.

This has the following advantages over the microwave or stovetop:

1. You can't forget about it and either overheat the water or wait too long and have it cool off too much (great for forgetful people like me)
2. It is much lower effort since it handles the process automatically
3. When you want to do another batch it is easy to heat back to the same point repeatedly or increase the temperature by small amounts as desired.
4. You don't need an external thermometer since the kettle has one built in

1. Efficiency

A microwave first converts the electric energy to microwaves, then send those waves into the chamber and there they are absorbed by an item inside. All of those processes are lossy, that is some of the energy is wasted as heat not in the food. You can hear a microwave fan that cools the electronics inside. That is the heat that's blown out into surrounding air and not into your water.

An electric kettle on the other hand, is a simple resistive heater. It means that all the energy going in gets converted into heat where you need it, that is on the heating element submerged in water.

This means that microwave will use more electricity to heat same amount of water to same temperature:

The magazine’s researchers discovered that an electric burner uses about 25 percent less electricity than a microwave in boiling a cup of water.

2. Power

Microwaves usually top at 800W range, while for a kettle, 800W is rather small one, a 1500W is common. This means that safe to assume that typical kettle will heat same amount of water in half the time.

Those are 2 biggest reasons why kettle is obviously better. There are many others, some mentioned here, although they are less prominent and often depend on your personal style or beliefs.

But - there are also reasons why sometimes it's better to use microwave:

1. People tend to overfill the kettle. Even when they need just a cup, they tend to fill it to the max (let's say 4 cups). This means that heating of remaining 3 cups is wasted, as this water will be left standing and cool down. That drops the efficiency to abyssal 25%! If you can't judge the amount of water before pouring it into kettle, then microwave can be more energy efficient after all.
2. Need of a certain temperature below boiling point. If you need water eg for melting chocolate, and you need it at certain temperature, then it's much easier to do it in a microwave. You can learn that eg 1 cup at 800W and 25 seconds yields the exact temperature you need. This is repeatable. Trying to do it in a kettle would require you watching it with a stopwatch, much less convenient and prone to errors.

Take into account steeping temperatures for tea and coffee, as well as the brand of hot chocolate you're using, as in its melting point and solubility in water/milk/cream. (TLDR How hot do you need the water to be and how much water do you need to heat). Electronic kettles and a pot on the stove give you temperature control but 1-2 minutes in a microwave will get you boiling point water temps (212 Fahrenheit/100 centigrade), this is mainly because microwaves are built to heat water molecules in food, to heat/thaw the food. So your microwave will heat the water faster than the stovetop and electric kettle.

No-one mentioned that you are supposed to pour the hot water onto the target (tea-bag, usually, these days).

In addition to being easier to control the temperature, easy to know when the temperature is correct, and not heating the drinking receptacle, a kettle is also easier to pour out of.

If you heat the water in a microwave in the cup, do you then try to dip your teabag into the hot cup and hot water, or do you pour the water carefully from that cup into another one with the teabag in it? In which case, why not just use a kettle?

The electric kettle is specially used for boiling water, and his existence must have his own reason, and the speed of heating the electric kettle water is very fast, which can improve efficiency.