VS Code 1.6.0 and Greater
As mentioned by aloisdg below, editor.renderWhitespace is now an enum taking either none, boundary or all. To view all whitespaces:
"editor.renderWhitespace": "all",
Before VS Code 1.6.0
Before 1.6.0, you had to set editor.renderWhitespace to true:
"editor.renderWhitespace": true