There's a neat feature now in FontForge (see @thomasrutter's answer) that lets you visually see all the glyphs in a given range, which helps when determining which unicode ranges that your font supports.
After opening the font with FontForge, click
Font Info... >
Unicode Ranges. This will give you an overview of how many glyphs exist in a given range. For example, for me it says
Basic Latin U+0020-U+007E 95/95
General Punctuation U+2000-U+206F 85/111
meaning that I have all of the glyphs in Basic Latin (95/95) but only 85 out of the 111 glyphs in General Punctuation.
Now, to see which 85 glyphs, click the "General Punctuation" entry in the list. This should make your other window jump to and highlight them as seen in the image below.
With this, it's easy to click the individual glyphs and figure out what ranges your font supports. In my case, it's U+2000-200B, U+2010-2027, and U+202F-205F in General Punctuation.
You would then have to do this for each of the ranges in the list.