the advantages of sending nuclear waste to the sun:
1. only NATURAL way of destorying the waste quickly
2. it wont affect the sun at all, its hot enough to break the stuff down to neutrons and so on (instead of big atoms)
3. its better then poluting whatever rock we otherwise leave it on (be it earth, moon, or some asteroid) cause the rock can be used for other stuff
reasons shuttles/rockets will never be used for it:
1. too expensive. them rockets, and their fuel can be quite pricy
2. too dangerous. strapping nuclear waste to the front of what is essentialy a big pipebomb is not a prety concept.
3. very wastefull. the ships (as they are now) are equiped/designed to COME BACK to earth for the most part, or atleast do something usefull (like turn/slowdown) in space.
why i think a rail gun is much more likely.
1. cheaper. while the initial investment is prety hefty to build it (the limiting factor so far), the cost of use is actualy prety minimal compared to anything else.
2. safer. there is no explosions (controled or otherwise). if you picture it as a big circle rail (probably on a magnet field istead of metal rails) that is used to speed the thing up to speed and then launched off the circle via a "branch" of the rail, then there's a failsafe (just let it slow down going in a circle if you wanna abort) that's safer then the aborted countdowns of a shuttle (stop an explosion once it started as i understand it).
3. not as error prone. some prety big tragectory problems can be ignored cause the sun will pull in most anything not in a good orbit around it (which this cant be unless its launched the wrong way, like away from the sun)
4. the rail gun launch thing, if built big enough, can be used to launch other stuff into orbit, by changing the launch tragectory and speed a little. so satelites (weather around the earth or sun) can be launched without the expence of massive rockets.