If I had a TARDIS, you know the first thing I would do is go see what a supernova looks like up close. I've even tossed around the idea of a little fanfic... but Megan Argo beat me to it. She's a radio astronomer at the Curtin University of Technology in Australia, and she wrote up a cute and engaging account of The Doctor and Martha witnessing an unusual exploding star (an audio version of the tale is available too)
The cool thing is, the story she wrote is actually part of a real event: the explosion of supernova SN2007gr, the death of a massive star. 2007gr was a Type Ic supernova, which is a star much more massive than the Sun, but has lost the majority of its outer layers over time due to a super-stellar wind. The core is basically all that's left, and when it runs out of fuel it collapses and then explodes.
2007gr was seen to have gas screaming away from it at almost half the speed of light, far faster than is typical for an exploding star. That means that the gas was focused into twin beams, probably shaped that way by the material swirling around the newly-formed black hole at its heart that formed in milliseconds after the collapse. It wasn't strong enough to be a monumentally violent gamma-ray burst, but it instead a sort-of hybrid object, one part normal supernova and one part GRB. We've known for some time that there is a connection between the two objects, but the actual events are difficult to study because they're uncommon. Supernova 2007gr is a rare opportunity to study one in detail.
But not as much detail as we could see if we had a time machine. Oh Doctor, there are some many things you could show us. But, I suppose, most of the fun is in figuring it out for ourselves.
Image: SN: NASA/CXC/M.Weiss; TARDIS: BBC; composition: Megan Argo