Drama without too much noise or banding.

Any monochrome gradient will be subject to posterization. A pure monchrome gradient has only 255 different values, not 255 x 255 x255. In practice, the good thing is that if the gradient changes color slightly, that means that you are no longer working on 255 values, but on the full range of 8-bits. For instance, if you have a 16-bits color image you want to desaturate, you'll get a lot more available colors and tones by using a gradient map adjustment layer with a sepia tone with slightly different starting and ending HUES.
That's pure math, but if you want to play visually, here are my favorite tools.

1 - The histogram panel.
If you see gaps in the histogram, typically a 'comb' of many spikes, that means that you have less different values, which appears in posterization.
16-bits purists use this combing as a hard proof that your image is spoiled. They have the math right. But... as a matter of fact, combing appears before you can really notice it in the image. So, it's an excellent tool to test the various factors influencing banding. Set the histogram to the RGB option. I say, trust your eyes (well, if your display is not even 8-bits like TV sets, you'll see display artifacts...)

2- The Posterize effect.
The menu Filter >> Adjustments >> Posterize will let you create posterized versions of your image.
You can play with any values between two (threshold effect) or 255 max (even if you are in 16-bits).

3- Gradients (expecially black to white)


You can use all those at the same time: creating a black to white gradient and comparing the visual effect and the histogram.

Those are the visual tools, but the very simple math requires the good question: when are tones stretched?
To simplify, first draw a full linear black to white gradient (0 to 255) from left to right.
This shows the tones available for black and white printing. It's about the minimum range of tones your eyes can distinguish. A few users will already guess posterization, less so if 'dither' is checked.
Now add a Posterize... adjustment layer and play with the different values. (Note that as revealed by Sepiana, you can simply move the slider in the adjustment panel in PSE15...)
Also have a look at the histogram to see the 'combing'.

Stretching happens when you start with a less contrast image or gradient, such as 33% to 66% in the end tabs of the gradient tools. That's only one third of the possible output range. If you do an auto levels, your image will be stretched to 0 to 100, and posterization will become clearly visible. Having only a third of a 16-bits range would not be a problem.

About blue gradients (sky, of course!)
If you examine the three channels separately (would need another discusson), you'll see that the blue channel has a very narrow range of luminosity. It must be stretched more than green or red.

Blending modes:
The above tools should help you experiment.
Start with the 'Equalize' filter. It will deliberately create posterization and combing in some parts of the histogram.
Normal, inverse, llighten or darken don't create stretching: they only use the existing values.
The overlay types, multiply and similar use 'multiply' calculations. Beware (look at histogram).
The difference does not produce stretching... but generally its result is so flat that you have to stretch it via auto levels, a curve or a gradient map adjustment layer.
Another command which can stretch tones is obviously the 'contrast' one, as well as curves and levels.

Flat areas or details.
Our eyes are better at distinguishing edges and contours from flat areas. They are also better as seen differences in luminosity than color (hue).
Other tools to experiment: start with patterns with more or less details. Or select scenes with many flat surfaces, skies, urban architecture and others with only details (trees, textures...)
Check the limits where posterization can happen with the posterization effect. Sharpenining and dust removal never need 16-bits.
Staying in normal mode with many layers and masks does not create posterization (compositing).

Noise and dithering.
When you want to deal with existing posterization, you may add noise. That's what dithering does in the gradients. Compare the visual output with the histogram.

I hope the above will get you started. We all have different eyes aptitudes, some may see posterization better, but the guides above should help everybody. Remember that your work will be seen at best with your good calibrated monitor or printer, and in many cases with bad or small displays .