Is there a quick, easy, mess-free way to peel an orange (or grapefruit)?
Yes!

I just came across this, and it just so happens I could use an orange in a slaw I'm making right now.



Tablespoon Orange Hack


  Simply lop off the ends, make a small incision in the side and carefully open to reveal easy-to-eat orange segments. (Or as I prefer to call it, an orange caterpillar.)


I did it with my orange, and I had never peeled an orange so quickly:



Now if you just want to eat the orange, you can just pluck off the segments as you pop them into your mouth. If you roll the orange around a bit first, before you lop off the ends (as in @Barry Staes' answer), it should make it even easier and more pith-free.

But what was really neat is that by prepping it that way, I had a pretty close approximation to supremes, supremely easily! (haha)

With two passes of the knife and 5 seconds of picking at it, this is close enough for my slaw.



Just flatten the "caterpillar" as much as you can with one hand, and with one slice (or two), cut between the peel and the segments. I just picked away most of the remaining connective "stuff" and membranes with the knife and my fingers at this point, but a pair of kitchen shears would  make even quicker work out of it. It's not absolutely perfect, the traditional method of supreming seen here would be the way to go if you're gunning for a Michelin Star, but this is fine for more casual applications.

With the practice of a couple more oranges, I think I could go from an intact orange to "cheater" supremed segments in less than a minute (and without getting my hands sticky!)

Pretty cool.

BTW, there is a pretty neat animated GIF on the linked page that will make it all clear if the pictures don't.

EDIT:

It works even better with grapefruit!

Open the grapefruit just enough to use a pair of kitchen shears to cut away the the connective membrane from the "points" of the segments (what would be at the center of intact fruit). While you're at it, remove any seeds.





Be careful when you roll out the "caterpillar" to separate every segment and to put pressure evenly on all parts of the peel. The peel is a bit more prone to tearing than orange peel.



Since the membranes between the segments are more like leather and less fragile than the membranes on orange segments, they are very easy to pluck away. This makes the supremed segments perfect. This gives better results even than the traditional method of supreming!