Show whitespace characters in Visual Studio Code

UPDATE (June 2019)

For those willing to toggle whitespace characters using a keyboard shortcut, you can easily add a keybinding for that.

In the latest versions of Visual Studio Code there is now a user-friendly graphical interface (i.e. no need to type JSON data etc) for viewing and editing all the available keyboard shortcuts. It is still under

File > Preferences > Keyboard Shortcuts (or use Ctrl+K Ctrl+S)

There is also a search field to help quickly find (and filter) the desired keybindings. So now both adding new and editing the existing keybindings is much easier:

Toggling whitespace characters has no default keybinding so feel free to add one. Just press the + sign on the left side of the related line (or press Enter, or double click anywhere on that line) and enter the desired combination in the pop-up window.

And if the keybinding you have chosen is already used for some other action(s) there will be a convenient warning which you can click and observe what action(s) already use your chosen keybinding:

As you can see, everything is very intuitive and convenient.
Good job, Microsoft!

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Original (old) answer

For those willing to toggle whitespace characters using a keyboard shortcut, you can add a custom binding to the keybindings.json file (File > Preferences > Keyboard Shortcuts).

Example:

// Place your key bindings in this file to overwrite the defaults
[
    {
        "key": "ctrl+shift+i",
        "command": "editor.action.toggleRenderWhitespace"
    }
]

Here I have assigned a combination of Ctrl+Shift+i to toggle invisible characters, you may of course choose another combination.