One of the items that appears to be missing from the Snipping Tool is a delayed mode so you can capture a menu or dropdown list after it is expanded. Well it turns out you can do a delayed capture with the Snipping Tool even if it's not obvious how to trigger it.
First start the Snipping Tool. If it's in capture mode hit Cancel so the window looks like:
Next open the menu or drop down the list you want to capture then press the "Ctrl" + "Print Screen" buttons on your keyboard. This will trigger a new capture with the screen locked as it is so you can get an image of the menu.
In Windows 10 it looks like the delayed mode will become easier to do with an explicit UI button to trigger it (info from Neowin)