I am going to answer this question setting limits around the amount of control one can have of any situation - so this would apply to situations where you are shooting wild life, sports, and news events or street situations where you have no real control over your subject, just some level of control over your relationship to the subject. (If you can control the room then one shot should be enough)
Off the top, I need to say that there is (can be) a huge difference between "spraying and praying" and firing off a 3 second volley at 6 fps, though it's not always apparent to anyone but the photographer. To know the difference you have to pay attention to everything they did in the minutes/hours before the shutter was pressed. I shoot a lot of wild life, and birds primarily. When I get to a new location that I've never shot at before, I do my best to assess 1) where the birds will be coming from, 2) where they might be landing (if anywhere), and 3) where the sun is, and then do my best to stick myself as close as I can to somewhere between locations 1 & 2 and location 3. After that, it's prep time, setting the camera the way I want it (mode, ISO, (min) shutter speed, focus and metering mode), having the right lens on, deciding if a monopod is appropriate, and waiting. For birds in flight, it's scanning the potential flight path, camera in hand, and firing a volley when something finally shows. Yes, spraying and praying - but with the prep work I've stacked the deck and am more likely to get an answered prayer, because I'm getting a bird that's flying towards me, in good light, and that might possibly land with wings fully extended and feathers displayed, captured at a shutter speed that won't blur them.
Now, when the subject is a little more stationary, there will be fewer volleys, but the prep work remains the same. Knowing where to be is again just as critical. This is where knowing your subject comes into play. In a concert setting, knowing how and when one musician reacts to another will allow you to grab the great "guitar face" photo, or that smile on a lean in, instead of just a shot of a couple blokes with guitars. In a news setting, knowing that a real long lens from that empty corner over there will allow you get the shot of the politician getting out of the car rather candidly before they approach the mob of photographers positioned near the building entrance. For animals perched in a tree, knowing the slight movements that signal a stretch of their wings or a jump from a branch will aid you in getting the moment of jumping and not just something falling thru the air. But in every case, you're damn straight I'm in continuous mode and will squeeze out more than one shot.
For me, this type of study has come at the cost of a lot of wasted shutter clicks and wondering why I keep only 3% of shots from 2 hours walking around a reservoir. Truth is, a lot of the waste is knowing when not to bother raising the camera because you've already missed it, so just watch and file the moment away so you don't miss it next time.
