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...

Here’s a rough-and-ready comparison, from a guy who’s mounted at least a half-dozen cartridges with a now ancient Dennesen Soundtracker --

Conical styli are inexpensive and easy to set up for pretty good, easy on the ears sound but they come up short on detail, imaging and air.

Elliptical will give you more detail, imaging & air than spherical/conical, and they’re still reasonably easy to mount and align onto the arm.

Chakster’s list of advanced shapes can all get you close to nirvana but set up and alignment become critical. Stylus rake angle has to be right on, which means you have the tone arm properly leveled (which can be a ticklish process). This includes having to account for how much the tracking force might deflect the tracking angle. It the adjustment is off, say hello to distortion, particularly as you reach the dreaded inner grooves....

Proper alignment of advanced stylus profile is not so difficult, manufacturers always give us recommended tracking force, the rest is just the same as with any profile (azimuth, overhang, tonearm vta).

Tonearm without adjustable VTA is junk if it wasn’t designed for one dedicated cartridge.

Azimuth adjustment can be made with a proper headshell with adjustable azimuth and overhang.

Protractor like Dr. Feickert NG is a must have for proper alignments (Baerwald, Lofrgen, Stevenson).

Everything can be adjusted if your vision is fine without any special devices.

Don’t be afraid to use advanced profiles.