Bolt action is far superior to pump or lever action when shooting in the prone position. Pump action is unsuitable because you have to move your support hand to operate it, which is bad because that arm is more involved with stabilizing shooter and rifle than the trigger hand. Lever action is unsuitable because, although it does use the trigger arm to operate the action, you need space below the rifle to operate a lever action, which may be difficult depending on where it's resting. Bolt action uses the trigger hand to cycle the rifle and does not require any ground clearance.

Both of the other answers talk about the importance of prone fire, and I think it's worthwhile to delve into this a little bit.

The prevailing orthodoxy after the Civil War was infantry would now be fighting at vast ranges owing to the general adoption of rifling in firearms. General issue gunsights would have ranges listed out to 2000 yards, although this was more for volley fire than individual use. This changed after the Battle of Little Bighorn, as the army was reluctant to acknowledge Custer's bad tactics and instead placed the blame at the single-shot weapons the troops were armed with. The "Trapdoor" Springfield single-shot rifle needed to be replaced with a repeater, and the question was which type.

Remember the extreme value placed on accuracy and range at this time. A lever gun has trouble holding the new spitzer-type rounds, as these point-y bullets if placed in a magazine tube had a habit of setting the one in front off by striking its primer. Yet spitzer bullets were much longer range than other types. Additionally, because the magazine tube was under the barrel it meant as rounds were used up the harmonics of the rifle shifted slightly with every shot. Finally, as other posters have noted, the inability to fire from a prone position - the most accurate position - was considered a deal breaker. So the Krag–Jørgensen was adopted, and later the 1903 Springfield, both bolt action guns.

An interesting historical footnote to illustrate the priorities of that era: many early bolt action guns had what was called a "magazine cut off", which prevented the cycling of the bolt from picking up a round from the magazine. The idea was the troops would use the weapons as single shot rifles, and then when they were charged or needed a great volume of fire their commanding officers would give them the order to switch to magazines and they'd get a boost of firepower. This was thought to encourage accuracy, lower ammunition consumption, and allow officers to better organize effective fire.

So why did pump actions take over with shotguns? Well if bolt action can be seen as high accuracy / low rate of fire, pump is the opposite. You get the fastest manual cycling method, at the cost of accuracy. You don't even need to take your finger off the trigger, just move the hand supporting the gun up and down and you're ready to let off another shell.

Lever action is something of the middle child, and so struggles to find a place in an era of semi-automatics. They still see use as hunting and self defence vs. animals weapons though, such as the Marlin Guide gun.

Lever action was rejected by militaries because it makes use in the prone position very impractical. You would either have to rotate the rifle or actually lift it up, potentially exposing yourself.

I do not know why bolt actions became dominant other than they got an early start in Europe, and momentum.

I would challenge that pump action is the "standard" mechanism for shotguns (mine and many others are double barrel, or over-under in my case) but anyway my speculation (is that allowed here?) is that the pump action allows you to keep sight on a moving target like a bird while operating the action, while that is much harder while operating a bolt which puts your right hand right next to your cheek and probably in your field of view.

Followup question: in WWI, how did the tens of thousands of left handed soldiers in million man armies cope with bolt action rifles designed for righties? Did they make left handed versions? Did they just have to suck it up.?

None of them have "faded out". Different mechanisms are used for different applications. For hunting every action is still used regularly, it comes down to a matter of preference and what caliber bullet you want to be using for a particular game. Some prefer a lever action when goat or deer hunting because the speed, safety, and ease when walking for miles looking for game. Some prefer bolt action or semi auto when posted up in a tree stand or blind. Bird hunters will regularly use pump, semi, or even a two barreled approach. And then when it comes to range shooting everything going back to black powder is still used. The military and governments tend to use bolt action for sniper rifles because of consistency. They use semi auto or burst auto for infantry because of ease of use and training. Pump action is used for a few reasons but mainly due to ease of use and intimidation. Merely pumping a shotgun can diffuse a situation. So to sum up; gun enthusiasts still shoot everything invented since the 1500s, hunters will shoot anything that suits their needs, and governments tend to be practical about how they arm their military.

For rifles, bolt action is a simple, reliable mechanism that allows for consistent firing, so for weapons like hunting rifles and sniper rifles that don't need to fire shots very quickly, bolt action is preferable as it is less likely to jam or be fouled by mud, snow, or other grime.

Shotguns are usually pump action just because, again, it's a reliable form of loading, with the shell sizes required a different action than cartridge ammunition. Pump action is simply what works, and as a plus shotguns have become known for the distinct sound of the pump action, and people know to respect the power of a shotgun in close quarters.

Levers are often discouraged in Hunter Safety Courses as unsafe. Of course, many hunters still use lever actions, and they're also popular in Cowboy Action shows.

A bolt action has some inherent safety, in that a bolt that isn't fully seated down will be physically impossible to fire accidentally. In a well designed bolt action, even lifting the bolt a few millimeters will disconnect the internal mechanisms enough to prevent an accident. While there might be additional safety systems beyond the bolt itself, these can be considered bonuses on top of a naturally safe system.

The Mosin's safety is an impossibly tight thing that has only gotten worse with age, so hardly anybody bothers with it. This is not much of a concern, since leaving the bolt part way up works just as well.

Compare this to a lever action, where the equivalent to having the bolt up is to have the lever down. Like the bolt, the mechanisms are disconnected in this situation. However, dropping the rifle could slam the lever shut while also jostling the weapon enough to cause it to misfire. It's very unlikely that a bolt would slam shut in the same situation.

Thus, adding on other sorts of mechanical safeties becomes important for a lever. One such safety is a half-cock trigger. This is where you hold the hammer with your thumb, pull the trigger, and slowly advance the hammer forward until it gets to the middle position. At that point, the hammer is locked in, and won't fire until it's manually pulled back again. Trouble is, if your thumb slips while setting it, it can still impact a primer and potentially set it off. In other words, the act of setting the safety can be unsafe!

Thus, the simplicity and safety of the bolt action has been encouraged in many Hunter Safety Courses, though this has not stopped some hunters from their personal preference in levers. Cowboy Action competitions and shows will no doubt continue to use them for theatrical purposes.

Other posters in this thread have made good points, it's also worth considering the changing role of mounted troops (cavalry) over the course of the 19th century. The original Spencer repeating rifle was very much designed with the needs of a cavalryman in mind -- and a lever action is somewhat easier to cycle from horseback than a bolt action.