Liquid Oxygen really is wonderful stuff. It's a beautiful blue and it's delightfully paramagnetic. It's also very, very cold (not as cold as LN2!) and very, very oxygenated. Duh. This makes it great for oxidizing and freezing whatever you like.
I have seen things burn in LOX and it is quite spectacular. Because the fuel has such great access to the O2, it burns with great fury. For example, I once saw a donut lit on fire with an oxy-acetelyne torch (actually, oxy-ethyne, but who needs the IUPAC anyway?). The flaming pastry was then dropped into a big beaker of LOX. It burned with such intensity that it really wasn't safe to look at for extended periods (Shade 6 glasses would have been good). The flaming donut spun around in the beaker, belching lots of smoke, which smelled of donut shop.
The concept of the oxy-cigar torch bears a small similarity to that demonstration.
An oxy-cigar torch is made by dipping the end of a cigar in LOX, freezing it. Then, it's lit on fire and one exhales through the cigar. The resulting torch can cut sheet metal (I recommend shade 5 glasses).
The source for this write up was Tom Thompkins, Master Exhibit Developer at the Exploratorium and personal hero, who has really done this.