I typically use just Aperture Priority and Manual modes 99.9% of the time, maybe more. I have not used Programmed very often, since I learned photography before P mode was common. Here's how I see things, from my old-fashioned POV:
Aperture Priority makes a lot of sense with new, modern digitial cameras that have wide dynamic range, and 3-D, RGB color-aware metering, and distance-sensing metering tied to the AF system. This means that Nikons made since the mid-1980's can sense the color of objects,their distance, and their reflectivity levels. Using the matrix metering concept, which Nikon invented in the Nikon FA film camera, NEWER digitial SLRs have one only type of "film" in them, color positive, and thus exposure is best tuned to keep the highlights from blowing out, and then processing the image so the lowest tones, the shadows, and the lower mid-tones, look darkish, yet detailed. Using Sony sensors, and scene-aware dynamic range enhancement techniques when processing in-camera JPEGs, the new Nikon cameras (and other brands) can produce a Matrix-metered, Aperture-priority auto shot that can look quite good, right out of the camera. This is NOT the way early digital cameras worked!
I like A mode for a lot of shooting. It is versatile,and it works well with Matrix metering pattern. With 3-D Matrix metering, a modern Nikon will usually give a pretty good exposure, and will pick the right shutter speed for the Aperture value you've selected. Aperture controls the "type"of picture we want: closeups often at f/16 to f/8; portraits often at f/8 to f/4.5; shallow DOF shots at f/2.8 to f/4, and so on. Also, depends on the lens: some lenses, some scenarios, will almost ALL be shot at say, f/4, or f/3.5, all day long. So A mode works well for those. A mode and Matrix metering can work wonderfully for fast shooting.
One trick with A mode in Center-weighted metering mode is to press and hold the thumb EV Lock button in, and lock in a desired exposure. If you keep the EV lock and the AF lock separated, one control on the thumb button (EV Lock) and the focusing on a separate button (shutter release) then you can use the A mode and Center-weighted metering to increase or decrease exposure by pointing the center circle at brighter or darker areas, and once the desired exposure correction is achieved, you can press the thumb button to lock-in the exposure. Want a faster speed? Swing the camera up toward the sky; want a darker exposure, swing the camera down toward a darker area, and in both cases, press the EV lock button to hold the "compensated" speed value. This method works extremely well using Center-weighted Metering, and Nikon's traditional 60/40 light meter pattern, in which 60% of the meter's area of emphasis is located inside that scribed 12mm center circle on the viewfinder screen. This is an old, old way of using A-mode center-weighted metering.
Manual mode: I do not like to use Matrix metering when using manual exposure control. Instead, I prefer to use Nikon's center-weighted metering. One can adjust this now, to very strongly-center weighted, less strong, and regular. (It's in the set-up menu for the camera.) Manual exposure control mode will work well with Spot Metering as well. 100 percent of my flash shots are made in Manual mode. In controlled, static settings, Manual mode keeps shots consistent. When shooting flash shots and changing lenses, the speed stays the same, and the Aperture value set on the body changes to the same f/stop the previous lens was set to: a very valuable feature when using studio flash units!
P mode is useful when you want to be able to SHIFT the f/stop and the shutter speed together, in lock step, to experiment with DOF effects, or motion effects, as you shoot quickly. This is, as I see it, the best use of P mode: moving from say wide-open at 1/1000 second, and then down, to slower speeds, or to smaller f/stops, like say f/4,then f/5.6 then f/8 then f/11. With motorized cameras and no film costs, this is more useful than it used to be. P mode can be used to great advantage by just rolling your thumb along the control wheel while shooting a sequence, and you can move from frozen motion, down to slightly blurred movement, and at the very far end, to blurry, panning-speed type shutter spoeeds and f/stop pairings. P mode allows you to experiment with DOF effects on individual scenes, and to do it very rapidly, and to come up with a variety of looks on things like floral close-ups, or scenic shots, or shots in the mid-distance with a long zoom, where the DOF can really affect how the foreground appears. P can be a good "experimental" mode.
S-Mode: only when the speed is **the** most critical parameter. Helicopters, propeller aircraft, and panning shots, and striving for specific motion-stopping ability, this is where S-mode becomes useful.
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You can now add Automatic ISO adjustment to the above ways of metering and exposing.