How can I find out a file's MIME type (Content-Type)?

Use file. Examples:

> file --mime-type image.png
image.png: image/png

> file -b --mime-type image.png
image/png

> file -i FILE_NAME
image.png: image/png; charset=binary

one of the other tool (besides file) you can use is xdg-mime

eg xdg-mime query filetype <file>

if you have yum,

yum install xdg-utils.noarch

An example comparison of xdg-mime and file on a Subrip(subtitles) file

$ xdg-mime query filetype subtitles.srt
application/x-subrip

$ file --mime-type subtitles.srt
subtitles.srt: text/plain

in the above file only show it as plain text.

file version < 5 : file -i -b /path/to/file
file version >=5 : file --mime-type -b /path/to/file