Stylus shape?

The shape of the stylus related not only to the better sound, but also to less record wear and longer life-span of the stylus itself. 

1) Conventional stylus is Spherical/Conical and Elliptical.

2) Advanced styli are: Hyper Elliptical, Shibata, Line Contact, Micro Line, Fine Line, Stereohedron, Micro Ridge, S.A.S., Van den Hul, Fritz Gyger, Replicant-100 .... and more.  

there are different names because of the different registered patent for each of them, in reality some of them are nearly identical to each other but registered under different patent. Look at this image. 

Advanced profiles are much more expensive.

Dear @musicfan2349 :  "  what shape of stylus yields the best sound.."

the stylus shape is only one of several parts in the design and manufacture of a phono cartridge and per sé is important but not determinant  in the overall cartridge quality level performance.

Exist several carrtridges with elliptical stylus shape that performs really good.

The build quality level in a cartridge is more important as it's too the analog rig that surrounds the cartridge and your ability to make the rigth overall cartridge/tonearm set up. The more advanced stylus shape needs a lot more effort and knowledge levels to make that set up in the rigth way where the cartridge can shows its real quality level performance .

Regards and enjoy the MUSIC NOT DISTORTIONS,
R.

@rauliruegas mentions setup. +1, the more extreme the styli profile the more effort to acheive its potential.
The designer and builder in such handmade transducers contribute a lot to the differences in sound so much so that bench racing by virtue of stylus shape or materials is difficult...