I have the 2011 version of the City Mini (which is the same as the 2012). I have several diaper bags, and the only one that fits underneath easily is a very small Land's End Little Tripper diaper bag. But, that bag is so small that I can hardly fit anything in it, so that doesn't work for trips of more than 2 hours. I have a Skip Hop Duo which can fit underneath, but only if it doesn't have much in it. Even then, it's hard to wrestle underneath and not worth the hassle.
The City Mini is a very stable stroller, so I don't have an issue with hanging a diaper bag on the handle. My favorite diaper bag (a Skip Hop Studio) is very easy to attach to the handle using the connector thingies. The Skip Hop Duo is easy to throw over the handle using the long strap. You can really attach just about any diaper bag to the handle using a set of Mommy Hooks.