[Tex/LaTex] What are all the font styles I can use in math mode

Question

As can be seen in above picture, there are lots of fonts or font styles that can be used in math mode.

What are all the fonts or font styles I can use in math mode, and which packages are required in order to use them?

Let's assume I'm using standard Computer Modern or Latin Modern (lmodern) fonts.

Edit: It'd be great to see examples of what the different styles look like, probably via images.

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_{I'm not interested in a reproduction of the picture, which can be found at What type of font is this?; the picture is for illustration purposes only. Furthermore, this is supposed to become a one-catches-all question for questions like the beforementioned. If I actually were looking for these fonts, I wouldn't just lazily ask here but look e.g. at
The Not So Short Introduction to LaTeX2e, tables 3.14 and 6.4 first.)}

Answer 1

In texdoc symbols (the Comprehensive LaTeX Symbol List) you can find Table 316:

The footnotes are explained in the document. Table 327 will additionally explain bold math.

Oh and needless to say but if you were asking this question because you need more mathematical symbols, the Comprehensive List is just your document. Greek variants, Hebrew, Tables 139 to 147 are letter-like symbols ... you'll probably never run out of symbols again.

Edit: After reading this answer https://tex.stackexchange.com/a/60018/13450 I feel the need to also point to the mathalfa package that is referenced there, providing even more alternatives than shown above.

Edit 2: This answer seems so popular that I decided to include the mathalfa table as an image as well. This is taken from the mathalfa documentation and some of the fonts are commercial or need to be installed from external sources. See the documentation for more information if you consider using any of these fonts. Warning, very long table ahead (stitched together from a multi-page table).

Edit 3: With this thread being so popular for reasons I don't entirely understand, I feel compelled to say that there is rarely a point in using more styles than regular, bold, italic, script/calligraphic (I wouldn't even mix those) and blackboard bold. What these tables really show are typefaces you can use for these styles, not a huge number of styles (which would be pointless and ugly anyway). If, however, you are just searching for math fonts to go with your main font, the overview you probably actually want before even consulting these tables is the list of math fonts on the LaTeX Font Catalogue.