How do I find out if a PNG is PNG-8 or PNG-24?

32-bit PNG (8 bit per channel - w/ alpha, also called "24-bit PNG with 8-bits of alpha" or "PNG24 with alpha"):

$ file zyne_zyne.png: 
PNG image data, 64 x 64, 8-bit/color RGBA, non-interlaced

24-bit PNG (8 bit per channel - no alpha channel)

$ file 24.png
PNG image data, 1060 x 821, 8-bit/color RGB, non-interlaced

8-bit PNG (256 colors):

$ file 8bit.png
PNG image data, 1060 x 821, 8-bit colormap, non-interlaced

The file command use in other answers is sometimes inconsistent. For example:

───────────────────────────────────────────────────────────────────────────────
rick@dell:~/Pictures$ file "256 colors.jpg"
256 colors.jpg: JPEG image data, JFIF standard 1.01, resolution (DPI), density 96x96, segment length 16, baseline, precision 8, 800x800, frames 3
───────────────────────────────────────────────────────────────────────────────
rick@dell:~/Pictures$ file "vnStat.png"
vnStat.png: PNG image data, 410 x 121, 8-bit/color RGB, non-interlaced

The first instance is likely a camera picture and the second instance I took using a screen capture.

The identify command, which my answer is based on, appears to work correctly in all instances. For example testing on the first instance above:

rick@dell:~/Pictures$ identify "256 colors.jpg"
256 colors.jpg JPEG 800x800 800x800+0+0 8-bit sRGB 45KB 0.000u 0:00.000

I have ImageMagick installed which allows me to use:

$ identify -verbose vnStat.png
Image: vnStat.png
  Format: PNG (Portable Network Graphics)
  Mime type: image/png
  Class: DirectClass
  Geometry: 410x121+0+0
  Units: Undefined
  Type: TrueColor
  Endianess: Undefined
  Colorspace: sRGB
  Depth: 8-bit
  Channel depth:
    red: 8-bit
    green: 8-bit
    blue: 8-bit
  Channel statistics:
    Pixels: 49610
    Red:
      min: 0 (0)
      max: 255 (1)
      mean: 12.0814 (0.0473779)
      standard deviation: 46.7032 (0.18315)
      kurtosis: 18.8998
      skewness: 4.49862
    Green:
      min: 0 (0)
      max: 255 (1)
      mean: 21.1804 (0.0830606)
      standard deviation: 57.5336 (0.225622)
      kurtosis: 6.68226
      skewness: 2.82613
    Blue:
      min: 0 (0)
      max: 255 (1)
      mean: 14.2606 (0.0559237)
      standard deviation: 50.2969 (0.197243)
      kurtosis: 13.5573
      skewness: 3.85914
  Image statistics:
    Overall:
      min: 0 (0)
      max: 255 (1)
      mean: 15.8408 (0.0621207)
      standard deviation: 51.7078 (0.202776)
      kurtosis: 11.8528
      skewness: 3.62638
  Rendering intent: Perceptual
  Gamma: 0.454545
  Chromaticity:
    red primary: (0.64,0.33)
    green primary: (0.3,0.6)
    blue primary: (0.15,0.06)
    white point: (0.3127,0.329)
  Background color: white
  Border color: srgb(223,223,223)
  Matte color: grey74
  Transparent color: black
  Interlace: None
  Intensity: Undefined
  Compose: Over
  Page geometry: 410x121+0+0
  Dispose: Undefined
  Iterations: 0
  Compression: Zip
  Orientation: Undefined
  Properties:
    date:create: 2017-01-24T20:04:50-07:00
    date:modify: 2017-01-24T20:04:50-07:00
    png:IHDR.bit-depth-orig: 8
    png:IHDR.bit_depth: 8
    png:IHDR.color-type-orig: 2
    png:IHDR.color_type: 2 (Truecolor)
    png:IHDR.interlace_method: 0 (Not interlaced)
    png:IHDR.width,height: 410, 121
    png:sRGB: intent=0 (Perceptual Intent)
    png:text: 1 tEXt/zTXt/iTXt chunks were found
    signature: e9f31b79da6ce46cdea5da21ae8648496faa181b0621098aa2dbbdff4a9a4502
    Software: gnome-screenshot
  Artifacts:
    filename: vnStat.png
    verbose: true
  Tainted: False
  Filesize: 14.7KB
  Number pixels: 49.6K
  Pixels per second: 0B
  User time: 0.000u
  Elapsed time: 0:01.000
  Version: ImageMagick 6.8.9-9 Q16 x86_64 2017-07-31 http://www.imagemagick.org

For a summary report omit all parameters:

$ identify vnStat.png
vnStat.png PNG 410x121 410x121+0+0 8-bit sRGB 14.7KB 0.000u 0:00.000

Replace vnStat.png with your own .png file name and preface it with the path if necessary.

To understand PNG-24 vs PNG-32 this answer from SuperUser summarizes it:

If your image is 24 bit you will see:

Channel depth:
  red: 8-bit
  green: 8-bit
  blue: 8-bit

If your PNG image is 32 bit you will see:

Channel depth:
  red: 8-bit
  green: 8-bit
  blue: 8-bit
  alpha: 8-bit

However this answer is not complete because 8-bit color also shows 8 bits for Red, Green and Blue, only a color palette is used to limit the number of selections (like in good old Windows 2.0).

Examine these three pictures from Wikipedia Depth Article:

File Name: 8_bit.png

Using:

identify -verbose  8_bit.png | grep colors
    png:PLTE.number_colors: 249

We see there are 249 colors used. The maximum possible with 8 bits is 256 colors.

File Name: 4_bit.png

Using:

identify -verbose  4_bit.png | grep colors
    png:PLTE.number_colors: 16

We see there are 16 colors used which is the maximum color combinations possible with 4 bits.

File Name: 2_bit.png

Using:

$ identify -verbose  2_bit.png | grep colors
    png:PLTE.number_colors: 4

We see there are 4 colors used which is the maximum color combinations possible with 2 bits.