To run grunt in debug, you need to pass the grunt script to node explicitly:

node-debug $(which grunt) task

and put a debugger; line in your task. node-inspector will then open a browser with debugging tools.

Edit 28 Feb 2014

node-inspector has added the command node-debug, which launches node in a --debug state and opens the browser to the node-inspector page, stopping when it hits the first debugger line or set breakpoint.

Edit 30 January 2015

On Windows, things are a touch more complicated. See the answer from @e.gluhotorenko for instructions.