Experimentally, taking this picture, and saving four JPEGs:

• one with full color
• one with dulled colors (saturation=50%)
• one with all colors reduced to grayscale (saturation=0%)
• one which is really single-channel grayscale (not shown below)

The single grayscale one is easily identified:

• the file utility reports only one component (three in the others)
• ImageMagick's identify command reports a Gray colorspace (sRGB for the others)

For the other three, a main differentiator is to count the unique colors, by design a grayscale JPEG image (whether true grayscale or RGB image with R=G=B) can have at most 256 colors when others are usually in the hundred thousands[*]. Using identify

$ for f in *.jpg ; do printf "%14s: %6d\n" $f $(identify -format "%k" $f) ; done
Chroma-000.jpg:    256
Chroma-050.jpg: 126123
Chroma-100.jpg: 345665
 Grayscale.jpg:    256

Using this it is easy (at least on OSX or Linux) to build a small script to move files depending on color count, for instance:

find . -name '*.jpg' -exec bash -c 'count=$(identify -format "%k" $1) ; [[ $count -le 256 ]] && exit 0 ; exit 1 ;' _ {} \;  -print

only print the names of the grayscale picture, and something like

find . -name '*.jpg' -exec bash -c 'count=$(identify -format "%k" $1) ; [[ $count -le 256 ]] && exit 0 ; exit 1 ;' _ {} \;  -exec mv -t /path/to/dir {} +

will move them to directory /path/to/dir.

[*] This may not be true for formats with high bit depths, for instance a grayscale 16-bit PNG could have 65536 colors (but 16-bot PNG aren't that frequent and can be discriminated with the file command or else). This is still less than the colors in the average photo. This also couldn't be true for CGI, but then we are on the Photography site.